AWK, 97 bytes

{l<NF&&l=NF;x++;for(t=k=0;k++<NF;t+=$k)a[x][k]=$k;if(t>g){g=t;b=x}}END{for(;m++<l;)print a[b][m]}

Attempt This Online!

{                          # code block runs with each column
l<NF&&l=NF;                # keep track of longest column
x++;                       # number of columns           
for(t=k=0;k++<NF;          # for each digit
t+=$k)                     # add to temp total 
a[x][k]=$k;                # place in array
if(t>g){g=t;b=x}}          # update if new total is larger

END{                       # runs after all data processed.
for(;m++<l;)print a[b][m]} # print our column

Japt -h, 7 bytes

May be a bit too liberal with the ability to "print these numbers in any format you like".

z3 ñȬx

Try it or run all test cases

Vyxal, 4 bytes

∩⁽∑∴

Try it Online! It's a bit weird that this challenge overrides default I/O, but thankfully vyxal casts the ints to digit lists anyway.

∩    # Transpose (casting each int to digits)
 ⁽ ∴ # Largest column by
  ∑  # Sum

Husk, 11 bytes

S!oS€▲mΣTmd

Try it online!

Takes input of a list of integers as an argument and outputs the "largest column" as a list too.

Not sure how exactly this works, but here is a tentative explanation:

            # Implicit input
         md # Start by conversion of every integer into its digits
        T   # Transpose or zip the lists into the columns
      mΣ    # Sum up the columns
     ▲      # Find the maximum column sum
   S€       # Find the position of the column with the maximum sum
            # (Combinator S passes 2 args to €: max and list of sums)
S!          # Another S-combinator: take the nth item of 
  o     Tmd # ...the earlier calculated transposed list

Pyth - 9 bytes

esD_.TjRT

Test Suite.

Pyth, 5 8 bytes

esDsMM.T

Takes in input as a list of strings, outputs as a nothing-separated list of digits.

Try it online!

Explanation:

      .T  Transpose input to zip together corresponding columns
   sMM    Cast to digit lists
 sD       sort(D) by (s)um
e         take last element, implicitly print

Javascript (ES6), 108 103 100 bytes

It's a bit verbose and could probably be golfed some more with a different approach. I wish I could get rid of this .filter(n=>n).

Saved 5 bytes thanks to Neil
Saved 3 bytes thanks to edc65

l=>l.map(n=>[...n].map((d,x)=>(m=(s[x]=(s[x]|0)-d)<m?s[c=x]:m,d)),s=[m=0]).map(n=>n[c]).filter(n=>n)

Demo

let f =
l=>l.map(n=>[...n].map((d,x)=>(m=(s[x]=(s[x]|0)-d)<m?s[c=x]:m,d)),s=[m=0]).map(n=>n[c]).filter(n=>n)

console.log(f(["1", "11", "111", "1111"]).join`,`);          // --> [1, 1, 1, 1]
console.log(f(["1", "12", "123"]).join`,`);                  // --> [2, 2]
console.log(f(["987654321", "111"]).join`,`);                // --> [9, 1]
console.log(f(["111", "123456789"]).join`,`);                // --> [9]
console.log(f(["4", "8", "15", "16", "23", "42"]).join`,`);  // --> [4, 8, 1, 1, 2, 4]
console.log(f(["4", "8", "12", "26", "27", "38"]).join`,`);  // --> [2, 6, 7, 8]
console.log(f(["24", "53"]).join`,`);                        // --> [2, 5]
console.log(f(["12", "304", "506"]).join`,`);                // --> [4, 6]
console.log(f(["30", "285", "121"]).join`,`);                // --> [0, 8, 2]

JavaScript (ES6), 90

(l,x)=>[...l+0].map((t,i)=>l.map(n=>(n=n[i])&&(v-=n,t.push(n)),v=t=[])|v>=x||(x=v,r=t))&&r

f=(l,x)=>[...l+0].map((t,i)=>l.map(n=>(n=n[i])&&(v-=n,t.push(n)),v=t=[])|v>=x||(x=v,r=t))&&r

;[
 [[123, 7, 49, 681]       , [2,9,8]]
,[[25, 223, 302]          , [2, 2, 3]]
,[[1, 11, 111, 1111]      , [1, 1, 1, 1]]
,[[1, 12, 123]            , [2, 2]]
,[[987654321, 111]        , [9, 1]]
,[[111, 123456789]        , [9]]
,[[4, 8, 15, 16, 23, 42]  , [4, 8, 1, 1, 2, 4]]
,[[4, 8, 12, 26, 27, 38]  , [2, 6, 7, 8]]
,[[24, 53]                , [2, 5]]
,[[12, 304, 506]          , [4, 6]]
,[[30, 285, 121]          , [0, 8, 2]]]
.forEach(t=>{
  var i=t[0], o=t[1], r, ok
  i=i.map(x=>x+'') // convert i to a string list
  r=f(i) 
  ok = (r+'')==(o+'') // compare r and o as comma separated strings
  console.log(ok?'OK':'KO', i+' -> '+ r)
  
})

Perl, 49 48 bytes

Includes +1 for -p

Run with the input on STDIN, prints to STDOUT the column numbers prefixed by +

lcolumn.pl
123
7 
49 
681

lcolumn.pl:

#!/usr/bin/perl -p
s/./@;[@-].="+$&"/eg}{($_)=sort{eval"$b<=>$a"}@

Pyth, 11 bytes

h.MsZ.TmjdT

A program that takes input of a list of integers on STDIN and prints a list.

Try it online

How it works

h.MsZ.TmjdT  Program. Input: Q
        j T   Yield the base-10 representation, giving a list of digits
       m d   Map that over Q (implicit input fill)
     .T      Justified transpose, giving each column as a list
 .MsZ        Filter the above by maximum sum
h            First element of above
             Implicitly print

Ruby, 100 97 bytes

a=$<.map &:chomp
puts a.map(&:size).max.times.map{|i|a.map{|e|e[i]}.compact}.max_by{|e|eval e*?+}

Mathematica 82 bytes

This pads the digits of each number with x's to the right, transposes the matrix, removes the dummy x's, orders by sum of the digits and takes the greatest.

SortBy[#~Select~NumberQ&/@Transpose[PadRight[#,30,x]&/@IntegerDigits@#],Tr][[-1]]&

There ought to be some way to use the superscript-T form of Transpose to save a few bytes.

Haskell, 63 bytes

import Data.Lists
argmax sum.transpose.map(map(read.pure).show)

Usage example: argmax sum.transpose.map(map(read.pure).show) $ [12,304,506] -> [4,6].

How it works:

                     map                       -- for each number
                         map(read.pure).show   -- turn into list of digits
           transpose                           -- transpose the list of list
argmax sum                                     -- find the element with the
                                               -- largest sum

Jelly, 6 bytes

DZṚSÞṪ

Try it online!. This is a relatively straightforward implementation of the question.

D              Convert each number in the input to a list of digits, e.g.
               [353, 2247] -> [[3, 5, 3], [2, 2, 4, 7]]
 Z             Zip the lists together, e.g. [[3, 2], [5, 2], [3, 4], [7]]
  Ṛ            Reverse the list of lists so that first occurrences are now
               at the end, e.g. [[7], [3, 4], [5, 2], [3, 2]]
   SÞ          Sort by sum - this uses Python's sorted function, which is stable
               so equal elements end up in order of appearance, e.g.
               [[3, 2], [7], [3, 4], [5, 2]]
     Ṫ         Tail - get the last element, e.g. [5, 2]