Uiua, 40 38 bytes

≡₀⨬@ (+@!⌊×94⚂)↘¯1/⊂↯+₁:↥⊸≡⇌⇌⊞(˜∊0⊟)∩⇡

40 bytes version:

≡₀⨬@ (+@!⌊×94⚂)↘¯1/⊂↯+₁:⍜⊣⋅1≡⍜⊃⊢⊣∩⋅1˜↯0⊂

Explanation:

Argument format: [HEIGHT] [WIDTH] [RUNG#]
≡₀⨬@ (+@!⌊×94⚂)↘¯1/⊂↯+₁:↥⊸≡⇌⇌⊞(˜∊0⊟)∩⇡ 
                             ⊞(˜∊0⊟)∩⇡ => create a HxW array of 0s, and fill in 1s for the top and left side
                        ↥⊸≡⇌⇌          => flip vertically, and horizontally flip to fill the right side also

40 bytes version:

≡₀⨬@ (+@!⌊×94⚂)↘¯1/⊂↯+₁:⍜⊣⋅1≡⍜⊃⊢⊣∩⋅1˜↯0⊂
                        ⍜⊣⋅1≡⍜⊃⊢⊣∩⋅1˜↯0⊂ => create a HxW array of 0s, and fill in 1s for the sides and the bottom
               ↘¯1/⊂↯+₁:                 => create R+1 copies of that, and remove the extra bottom
≡₀⨬@ (+@!⌊×94⚂)                          => at the truthy idxs generate a printable ASCII char, otherwise a space

Try it online!

Vyxal, 20 bytes

›*‹⟑kP⁰ƈÞẇn¹Ḋ*1↳JǓ∑,

Try it Online!

Takes N, H, W

Explained

›*‹⟑kP⁰ƈÞẇn¹Ḋ*1↳JǓ∑,­⁡​‎‎⁡⁠⁡‏⁠‎⁡⁠⁢‏⁠‎⁡⁠⁣‏‏​⁡⁠⁡‌⁢​‎‎⁡⁠⁤‏‏​⁡⁠⁡‌⁣​‎‎⁡⁠⁢⁡‏⁠‎⁡⁠⁢⁢‏⁠‎⁡⁠⁢⁣‏⁠‎⁡⁠⁢⁤‏‏​⁡⁠⁡‌⁤​‎‎⁡⁠⁣⁡‏⁠‎⁡⁠⁣⁢‏‏​⁡⁠⁡‌⁢⁡​‎‎⁡⁠⁣⁣‏⁠‎⁡⁠⁣⁤‏⁠‎⁡⁠⁤⁡‏‏​⁡⁠⁡‌⁢⁢​‎‎⁡⁠⁤⁢‏‏​⁡⁠⁡‌⁢⁣​‎‎⁡⁠⁤⁣‏⁠‎⁡⁠⁤⁤‏‏​⁡⁠⁡‌⁢⁤​‎‎⁡⁠⁢⁡⁡‏‏​⁡⁠⁡‌⁣⁡​‎‎⁡⁠⁢⁡⁢‏‏​⁡⁠⁡‌⁣⁢​‎‎⁡⁠⁢⁡⁣‏⁠‎⁡⁠⁢⁡⁤‏‏​⁡⁠⁡‌­
›*‹                   # ‎⁡H*(N+1)-1 -> the total number of lines in the output
   ⟑                  # ‎⁢For each number n in the range [1, H*(N+1)-1]:
    kP⁰ƈ              # ‎⁣  Select N random characters from the set of printable ascii characters
        Þẇ            # ‎⁤  Take the first and last characters from that list, and push them under the remaining characters (i.e. push [first, last], characters[1:-1])
          n¹Ḋ         # ‎⁢⁡  Check whether n (from the original range) is divisible by H. Because the range is 1-indexed, this will nicely align with which lines need a full row.
             *        # ‎⁢⁢  Only keep the letters in the row if this is a line where a row should be. Replace with a list of empty strings otherwise.
              1↳      # ‎⁢⁣  Turn those empty strings into single spaces.
                J     # ‎⁢⁤  And merge the modified row with the original first and last characters. The top of the stack is now [first, last, row...]
                 Ǔ    # ‎⁣⁡  Rotate that left once to put the first character on the end, bookending the row. This works because the first and last characters of a row are guaranteed to always be present.
                  ∑,  # ‎⁣⁢  Concatenate that list into a single string and print with a newline (doesn't break lazy evaluation because this is an eagerly evaluated map).
💎

Created with the help of Luminespire.

Zsh, 90 bytes

r()jot -rc $1 ! ~ -s ''
s()repeat $2-1<<<${(r:$1-1:)$(r 1)}`r 1`
repeat $3 s $@&&r $1;s $@

Try it online! _95b

W=$1, H=$2, N=$3
Function r() asks jot for a list of x random numbers in the character range. Function s() draws the 2 sides of height H-1. The main script repeatedly calls s and r to draw the sides and rungs, N times. The final call to s $@ draws the bottom legs of the ladder.

Originally I tried to use code from "Do X without Y" with complicated logic but calling jot is simpler than doing math and ASCII conversions.

Operation Flashpoint scripting language, 643 624 bytes

f={l=["""","!","#","$","%","&","'","(",")","*","+",",","-",".","/","0","1","2","3","4","5","6","7","8","9",":",";","<","=",">","?","@","A","B","C","D","E","F","G","H","I","J","K","L","M","N","O","P","Q","R","S","T","U","V","W","X","Y","Z","[","\","]","^","_","`","a","b","c","d","e","f","g","h","i","j","k","l","m","n","o","p","q","r","s","t","u","v","w","x","y","z","{","|","}","~"];t=_this;w=t select 0;r={l select random 92};v="";s=v;i=2;while{i<w}do{i=i+1;v=v+" "};p={i=1;while{i<t select 1}do{i=i+1;s=s+call r+v+call r+"\n"}};k=0;call p;while{k<t select 2}do{k=k+1;i=0;while{i<w}do{i=i+1;s=s+call r};s=s+"\n";call p};s}

Ridiculously long because there is no way to create the characters from the character codes.

Call with:

hint ([5, 3, 2] call f)

Output:

The ladder is extra chaotic because the font is not monospaced.

Unrolled:

f =
{
    l = ["""","!","#","$","%","&","'","(",")","*","+",",","-",".","/","0","1","2","3","4","5","6","7","8","9",":",";","<","=",">","?","@","A","B","C","D","E","F","G","H","I","J","K","L","M","N","O","P","Q","R","S","T","U","V","W","X","Y","Z","[","\","]","^","_","`","a","b","c","d","e","f","g","h","i","j","k","l","m","n","o","p","q","r","s","t","u","v","w","x","y","z","{","|","}","~"];

    t = _this;
    w = t select 0;

    r =
    {
        l select random 92
    };

    v = "";
    s = v;

    i = 2;
    while {i < w} do 
    {
        i = i + 1;
        v = v + " "
    };

    p =
    {
        i = 1;
        while {i < t select 1} do 
        {
            i = i + 1;
            s = s + call r + v + call r + "\n"
        }
    };

    k = 0;
    call p;
    while {k < t select 2} do 
    {
        k = k + 1;

        i = 0;
        while {i < w} do
        {
            i = i + 1;
            s = s + call r
        };
        s = s + "\n";

        call p
    };

    s
}

Japt -R, 42 41 40 37 34 28 25 bytes

Takes input in the order H,W,N.

;*°WÉ ÆJ²ùVXgJùU¹r,@EÅöÃé

Try it

R, 138 129 111 98 93 bytes

-13 bytes thanks to Neal Fultz!

-1 byte thanks to Robin Ryder

function(W,H,N){m=matrix(intToUtf8(32+sample(94,W*(h=H*N+H-1),T),T),h)
m[-H*1:N,3:W-1]=" "
m}

Try it online!

Anonymous function; returns the result as a matrix.

Thanks to that Word Grids question, I've been thinking about matrices a lot more than usual. I observed that the rungs are in those matrix rows that are a multiple of the step height H (R is 1-indexed), and that the rails are the first and last columns, 1 and W. So I create a matrix of random ASCII characters, and replace those letters that didn't match those criteria with spaces, and return the matrix. TIO link prints it out nicely.

Neal Fultz suggested a different indexing for the space characters, [-H*(1:N),3:W-1], which replaces all characters except for those in rows of multiples of H: -H*(1:N) and not on the edge, 3:W-1 <==> 2:(W-1).

R, 121 bytes

function(W,H,N)for(i in 1:(H*N+H-1)){for(j in 1:W)cat("if"(!(i%%H&j-1&j-W),sample(intToUtf8(33:126,T),1)," "))
cat("\n")}

Try it online!

An improvement over the original matrix-based approach I started with; it's the same algorithm but for loops are shorter than constructing and printing a matrix (but not if I don't print it!)

Ruby, 71 bytes

EDIT: Oops I thought this was a new challenge because of the recent edit to fix a typo lol. I'm still leaving this up though because there's no Ruby answer for it yet.

->w,h,n{(1..h*-~n-1).map{|i|[*?!..?~].sample(x=i%h>0?2:w)*(' '*(w-x))}}

Try it online!

Perl 6, 76 73 bytes

->\h,\n,\w{map {chrs (roll(w,1..94)Z*1,|($_%%h xx w-2),1)X+32},1..^h*n+h}

Try it online!

Takes (h, n, w) as arguments. Returns a list of strings.

Explanation:

-> \h, \n, \w {  # Block taking arguments h, n, w
    map {
        # String from codepoints
        chrs
             # Generate w random numbers between 1 and 94
             (roll(w, 1..94)
              # Set inner numbers on non-rungs to zero
              Z* 1, |($_%%h xx w-2), 1)
             # Add 32 to numbers
             X+ 32
    }
    # Map h*n+h-1 row numbers (1-indexed)
    1..^h*n+h
}

Powershell, 102 bytes

param($w,$h,$n)1..(++$n*$h-1)|%{$l=$_%$h
-join(1..$w|%{[char](32,(33..126|Random))[!$l-or$_-in1,$w]})}

Less golfed test script:

$f = {

param($w,$h,$n)
1..(++$n*$h-1)|%{       # for each lines of the ladder
    $l=$_%$h            # line number in a step
    -join(1..$w|%{      # make a line
        [char](32,(33..126|Random))[!$l-or$_-in1,$w]
    })                  # a random character if the line number in a step is a rung line or char position is 1 or width
                        # otherwise a space
}

}

&$f 5 3 2
&$f 3 2 2
&$f 12 4 5
&$f 20 5 3

Output:

0   {
H   S
']UxR
G   ]
3   t
q^R8O
q   y
t   J
U h
YQZ
_ i
3#D
I #
=          m
&          <
]          6
8nmuyw2'Y7%+
o          l
;          !
D          M
Fn[zGfT";RYt
@          B
$          e
z          @
@J[1|:-IS~y<
(          L
:          [
|          q
zBow0T0FnY8)
/          *
e          B
R          p
9{d2(RacBdRj
u          ~
`          l
J          h
v                  t
T                  -
v                  H
'                  Y
IS7{bx2&k@u7]o}>[Vq?
F                  U
?                  U
|                  Q
}                  T
:wv1wEfc6cS;430sigF|
<                  L
:                  }
*                  `
H                  =
L8k5Q/DQ=0XIUujK|c6|
j                  =
!                  p
V                  :
#                  w

MATL, 63 50 bytes

-13 bytes thanks to Luis Mendo

Q*qXJ*6Y2HY)wT3$ZrJ3G3$eJ3G&Ol5LZ(J:HG\~3GTX"!+g*c

Try it online!

I'm still new to golfing in MATL (and I'm not super good at MATLAB for that matter), so I know this probably isn't close to optimal. Tips are welcome. Takes input in order N,H,W.

Here we go:

Q*qXJ                     # compute H*(N+1)-1, store as J
     *                    # multiply by W
      6Y2HY)              # push printable ASCII
            wT3$Zr        # sample uniformly with replacement
                  J3G3$e  # reshape to make a matrix of the appropriate shape.

We now have a matrix of random char.

J3G                       # push J,W
   &O                     # zero matrix, J x W
     l5LZ(                # assign 1 to first and last columns

Now there's also a logical matrix for the rails.

J:                        # push J, range, so 1...J
  HG                      # take second input (H)
    \~                    # mod and bool negate (so it's 1 for rows of multiples of H)
      3GTX"!              # repmat and transpose so we have 1's for rungs

Now we have 3 matrices on the stack:

• Top: 0 for non-rung, 1 otherwise
• Middle: 0 for non-rail, 1 otherwise
• Bottom: random characters, -20

So we do the following:

+                         # add the top two matrices.
 g                        # convert to logical so 0->0, nonzero->1
   *                      # elementwise multiply
    c                     # convert to char, implicit output (0 -> space).

Haskell, 226 220 211 190 bytes

import System.Random
a=mapM id
b=(putStr.unlines=<<).a
c=randomRIO('!','~')
r w=a$c<$[1..w]
s w=a$c:(return ' '<$[3..w])++[c]
(w#h)0=b$s w<$[2..h]
(w#h)n=do{b$(s w<$[2..h])++[r w];(w#h)$n-1}

Try it online!

Saved 9 bytes thanks to Laikoni

Saved 21 bytes thanks to wchargin

Should be golfable (b$(s w)<$[2..h] and b$((s w)<$[2..h])++[r w]). I don't feel comfortable with IO and golfing.

Python 2, 142 bytes

lambda w,h,n,e=lambda:chr(randint(33,126)):[e()+[eval(("e()+"*(w-2))[:-1])," "*(w-2)][-~i%h>0]+e()for i in range(h*-~n-1)]
from random import*

Try it online!

Saved bytes thanks to ovs!

QBIC, 76 bytes

[:*:+b-1|G=chr$(_r33,126|)~a%b\[:-2|G=G+@ `]][e|G=G+chr$(_r33,126|)]?_sG,1,e

Explanation

[                       FOR a = 1 TO
 :                         input 1 (Height, now in var b)
  *                        times
   :                       input 2 (# of rungs, now in var c)
    +b-1|                  plus one bottom rung without crossbar
G=chr$(_r33,126|)       Assign to G a random char (_r is the RAND() function, chr$() is BASIC's num-to-char)
~a%b|                   IF we are not at a crossbar (the modulo returns anything but 0), 
  [:-2|G=G+@ `            add to G x spaces, where x is width (input 3, now 'e') - 2
                        Note that G is now either 'X' or 'X   '  (for rnd char X and W=5)
]]                      Close the spacing FOR, close the IF 
[e|                     FOR f = 1 to width
  G=G+chr$(_r33,126|)]     Append to G a rnd char
                        G is now either 'XXXXXX'  or 'X   XXXXX' (for rnd char X and W=5)
?_sG,1,e                PRINT the first w characters of G, and on to the next line

Sample run

Command line: 3 2 5
N   F
M   `
Bj#=y
!   (
S   N
q(.Ho
%   7
g   ,

JavaScript (ES6), 129 117 bytes

Unfortunately, while I was in the process of golfing this down, Arnauld beat me to a similar but shorter solution. By combining our 2 solutions, this can be 113 bytes

Includes a trailing newline.

(w,h,n)=>(g=c=>l?(c++<w?c%w>1&&l%h?` `:String.fromCharCode(94*Math.random()+33|0):`
`)+g(c>w?!--l:c):``)(0,l=h*++n-1)

Try it

o.innerText=(f=
(w,h,n)=>(g=c=>l?(c++<w?c%w>1&&l%h?` `:String.fromCharCode(94*Math.random()+33|0):`
`)+g(c>w?!--l:c):``)(0,l=h*++n-1)
)(i.value=5,j.value=3,k.value=2);oninput=_=>o.innerText=f(+i.value,+j.value,+k.value)

label,input{font-family:sans-serif;font-size:14px;height:20px;line-height:20px;vertical-align:middle}input{margin:0 5px 0 0;padding:0 0 0 5px;width:100px;}

<label for=i>W: </label><input id=i min=3 type=number><label for=j>H: </label><input id=j min=2 type=number><label for=k>N: </label><input id=k min=2 type=number><pre id=o>

C (gcc), 141 131 114 109 107 bytes

Should be able to golf this down quite a bit...

i,j,c;f(w,h,n){for(i=1;i<h*n+h;i+=j==w)printf(i%h?i++,j=0,"%c%*c\n":"%c",++j^w?c^8:10,w-2,c=33+rand()%94);}

Try it online!

JavaScript (ES6), 117 115 bytes

A recursive function building the output character by character.

"Look Ma, no literal line-feed!"

(w,h,n)=>(g=x=>y<h*n+h-1?String.fromCharCode(x++<w?x%w>1&&-~y%h?32:Math.random()*94+33|0:10)+g(x>w?!++y:x):'')(y=0)

Demo

let f =

(w,h,n)=>(g=x=>y<h*n+h-1?String.fromCharCode(x++<w?x%w>1&&-~y%h?32:Math.random()*94+33|0:10)+g(x>w?!++y:x):'')(y=0)

O.innerText = f(5,3,4)

<pre id=O></pre>

Charcoal, 34 32 bytes

Ｅ…¹×⁺¹ＮＮ⪫ＥＩζ⎇∧﹪ιＩη﹪λ⁻Ｉζ¹ §⮌γ‽⁹⁴ω

Try it online! Takes input in the order N, H, W. Verbose approximation (Plus(InputNumber(), 1) is currently broken on TIO). Explanation:

Ｅ…¹×⁺¹ＮＮ

Map over the range 1..H*(N+1). This means that the rungs appear when i is a multiple of H.

⪫

Join the result of:

ＥＩζ

mapping over the implicit range 0..W:

⎇∧﹪ιＩη﹪λ⁻Ｉζ¹ 

if the column is not 0 or W-1 and the row is not a multiple of H then output a space;

§⮌γ‽⁹⁴

otherwise, take the predefined ASCII character variable, reverse it (putting the space in 94th place), and print a random character from what is now the first 94. (Because Slice sucks.)

ω

Join using the empty string. Final result is implicitly printed.

Java 8, 203 188 168 133 132 130 128 126 bytes

W->H->N->{for(double i=0,j,q;++i<H*N+H;)for(j=W,q=10;j-->=0;q=i%H*j<1|j>W-2?33+Math.random()*94:32)System.out.print((char)q);}

Try it online!

05AB1E, 29 bytes

Input taken in the order N, H, W

>*GNUžQ¦©.RIÍFð®.R«X²Öè}®.RJ,

Try it online!

Explanation

>*G                              # for N in [1 ... (N+1)*H)-1] do:
   NU                            # store N in variable X
     žQ                          # push a string of printable ascii
       ¦©                        # remove the first (space) and save a copy in register
         .R                      # pick a random character
           IÍF                   # W-2 times do:
              ð                  # push a space
               ®.R               # push a random ascii character
                  «              # concatenate
                   X²Ö           # push X % H == 0
                      è          # index into the string of <space><random_char> with this
                       }         # end inner loop
                        ®.R      # push a random ascii character
                           J,    # join everything to a string and print

Jelly,  24 23  22 bytes

«þỊoU$ẋ⁵‘¤Ḋ×94X€€+32ỌY

A full program taking the three arguments W, H, N and printing the result.

Try it online!

How?

Builds a 2d array mask of a single rung and its vertical sections below, repeats it N+1times and removes the top rung then places random characters or spaces depending upon the mask value.

«þỊoU$ẋ⁵‘¤Ḋ×94X€€+32ỌY - Main link: W, H (N is third input / 5th command line argument)
 þ                     - outer product (build a table using):
«                      -  minimum
                       -  ...note: þ implicitly builds ranges of W and H prior
  Ị                    - insignificant? (abs(z)<=1) - yields a W by H 2-d array,
                       -   all zeros except the left and top edges which are 1s
     $                 - last two links as a monad:
    U                  -   upend (reverse each row)
   o                   -   or (vectorises) - giving us our |‾| shape of 1s
         ¤             - nilad followed by link(s) as a nilad:
       ⁵               -   5th command line argument, N
        ‘              -   increment -> N+1
      ẋ                - repeat list - giving us our ladder-mask plus a top rung)
          Ḋ            - dequeue - remove the top rung
            94         - literal ninety-four
           ×           - multiply (vectorises) - replace the 1s with 94s
              X€€      - random for €ach for €ach - 0 -> 0; 94 -> random integer in [1,94]
                  32   - literal thirty-two
                 +     - add (vectorises) - 0 -> 32; random integers now from [33,126]
                    Ọ  - character from ordinal (vectorises)
                     Y - join with newlines
                       - implicit print

C, 95 bytes

f(w,h,n,i){++w;for(i=0;i++<w*~(h*~n);)putchar(i%w?~-i%w%(w-2)*((i/w+1)%h)?32:33+rand()%94:10);}

SOGL V0.12, 32 31 bytes

 ~ΔkψR
I{e{R}¶bH{Re⁾⌡@R¶}}¹∑e⌡k

Try it Here!

Input in the order N, W, H.

Explanation:

 ~ΔkψR

     R  a function named "R", pushes a random character:
 ~       push "~"
  Δ      get the charasters from " " to "~"
   k     remove the 1st character
    ψ    choose a random character from that

I{e{R}¶bH{Re⁾⌡@R¶}}¹∑e⌡k  main program

I                         increase the 1st input - N
 {                }       N times do
  e                         push the variable e, which is here initialised to the next input - W
   { }                      E times do
    R                         execute R
      ¶                     push a newline
       bH                   push b-1, where b is initialised to the next input - H
         {       }          B-1 times do
          R                   execute R
           e⁾                 push e-2 aka width-2
             ⌡@               push that many spaces
               R              execute R
                ¶             push a newline
                   ¹∑     join the stack together
                     e⌡k  remove the first width characters

18 bytes without the random characters :/

Python 2, 114 bytes

lambda w,h,n:[[chr(32+randint(1,94)*(x%~-w*(y%h)<1))for x in range(w)]for y in range(1,h*-~n)]
from random import*

Try it online!

JavaScript (ES6), 144 bytes

(w,h,n)=>Array(n+1).fill(("#".padEnd(w-1)+`#
`).repeat(h-1)).join("#".repeat(w)+`
`).replace(/#/g,_=>String.fromCharCode(33+Math.random()*94|0))

Creates the ladder out of # characters then replaces each one with a random ASCII char.

Test Snippet

let f=
(w,h,n)=>Array(n+1).fill(("#".padEnd(w-1)+`#
`).repeat(h-1)).join("#".repeat(w)+`
`).replace(/#/g,_=>String.fromCharCode(33+Math.random()*94|0))

;(W.oninput=H.oninput=N.oninput=B.onclick=_=>{let vals=[W.value,H.value,N.value].map(eval);D.innerText=vals.join`, `;O.innerText=f(...vals);})()

W: <input id=W type=range min=3 max=25 value=12><br>
H: <input id=H type=range min=2 max=25 value=5><br>
N: <input id=N type=range min=2 max=10 value=2><br>
<button id=B>Rerun</button> <code id=D></code>
<pre id=O></pre>

PowerShell, 132 124 bytes

param($w,$h,$n)-join([char[]]((($a=('#'+' '*($w-2)+"#`n")*--$h)+'#'*$w+"`n")*$n+$a)|%{($_,[char](33..126|Random))[$_-eq35]})

Try it online!

We construct a ladder composed of only # first (example), then loop |%{...} through each character and if it's -equal to 35, we pull out a new Random character from the appropriate range. Otherwise we output (i.e., either a space or newline).

Pyth, 33 bytes

VhEjtW!Nmsm?&d}kr1tQ\ Or\!C127Qvz

Try it online: Demonstration

Perl 5, 81 bytes

80 bytes code + 1 for -p.

/ \d+ /;$_=(($}=(_.$"x($`-2)._.$/)x($&-1))._ x$`.$/)x$'.$};s/_/chr 33+rand 94/ge

Try it online!