R, 104 bytes

function(s){m=matrix(" ",l<-nchar(s),l)

m[cbind(1:l,sample(l,l,T))]=el(strsplit(s,""))

write(m,1,l,,"")}

Try it online!

Input as a string; writes to stdout.

Pyth - 9 bytes

Outputs list of lines.

.tm+*;OlQ



 .t                       Transpose, padding with spaces

  m      (Q implicit)     Map over input

   +    (d implicit)      Concatenate to loop var

    *                     String repeat

     ;                    This refers to the var replaced by loop var, which is d=" "

     O                    Random number less than

      lQ                  Length of input

Try it online.

APL (Dyalog Unicode), 16 bytes^SBCS

Anonymous tacit prefix function

⍉∘↑⊢↑¨⍨∘-∘?≢⍴1+≢

≢ length of string

1+ one added to that

≢⍴ "length" copies of that

∘? random integers in range 1…those, and then…

∘- negate, and then…

⊢↑¨⍨ take than many elements from each character, padding on the left with spaces

∘↑ mix list of strings into matrix, padding with spaces on the right

⍉ transpose

Try it online!

Ruby, 59 57 bytes

->a{a.map{|c|s=[' ']*z=a.size;[s[rand z]=c]+s}.transpose}

Try it online!

Inputs 1D, outputs 2D array of characters.

Japt, 8 bytes

-1 byte from @Shaggy

y_iUÊö ç

y_iUÊö ç        Full Program. Implicit input U

y_              transpose and map each row in U (Call it X)

  i             Insert at the beginning of X:

       ç        " " repeated ↓ many times

   UÊö          random integer in [0, length of U] 

                implicit transpose back and output

Try it online!

JavaScript (ES6), 72 bytes

Takes input as a list of characters. Returns a matrix of characters.

a=>a.map((_,y)=>a.map((c,x)=>Math.random()<.5|!a[y+1]?(a[x]=' ',c):' '))

Try it online!

J, 30 19 bytes

|:@,.]{.~"+_2-#?@##

Try it online!

Japt, 8 bytes

yÈùUÊö Ä

Try it

yÈùUÊö Ä     :Implicit input of string U

y            :Transpose

 È           :Pass each column through the following function and transpose back

  ù          :  Left pad with spaces to length

   UÊ        :    Length of U

     ö       :    Random number in the range [0,UÊ)

       Ä     :    Plus 1

PHP, 88 bytes

for($o='';$i<$l=strlen($argn);$o[$i+$l*rand(0,$l)]=$argn[$i++]);echo chunk_split($o,$l);

Try it online!

Input from STDIN, output to STDOUT. Run as:

$ echo Hello World|php -nF rain.php

   l  W    

  l        



 e      r d



H      o   



         l 





    o 

-1 byte (empty string instead of space) and +1 byte (err on side of rules) thx to @ASCII-only!

Python - 78 bytes

lambda s:map(None,*[(random.randrange(len(s))*' '+c).ljust(len(s))for c in s])

Jelly, 10 bytes

³LŻX⁶x;)z⁶

Try it online!

      )    | For each input character

³L         | Length of original input

  Ż        | 0..length

   X       | Random number from that list

    ⁶x     | That number of spaces

       ;   | Concatenate to the character

        z⁶ | Finally transpose with space as filler

Charcoal, 10 9 bytes

↑Ｅθ◧ι⊕‽Ｌθ

Try it online! Link is to verbose version of code. Edit: Saved 1 byte thanks to @ASCII-only. Explanation:

  θ         Input string

 Ｅ          Map over characters

        θ   Input string

       Ｌ    Length

      ‽     Random value

     ⊕      Incremented

    ι       Current character

   ◧        Padded to length

↑           Print rotated

As ASCII-only points out, you can move the letters randomly up instead of down for the same effect (except that there might be extra white space at the bottom rather than the top). Printing an array of characters upwards is equivalent to printing a string normally, so the padding then just offsets each character vertically by a random amount.