To stop asking for more user input after the word 'done' has been entered, you may simply put <STDIN> inside a loop containing also a conditional (for exit):

use strict; # never go by without it! ;)

rint "Enter your data and then it will print out in reverse order.\n";

 my @input;
 while (<STDIN>){
   chomp; 
   last if m/done/i;  # quit this loop if input 
                      # matches (case insensitive) the word 'done'.
   push @input, $_;   # otherwise, add the word to the list
 }

 # Use 'for' loop instead of while if you need to
 # access array elements via an index variable.
 for (my $currentline = 1; $currentline <= scalar @input; $currentline
+++)
 {
  my @words=split(/ /,$input[$currentline - 1]);
  #@words=reverse(@words);
  $input[$currentline - 1]=join(" ",@words,"\n");
 }

 @input=reverse(@input);
 print(@input);
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UPDATE: polished your code a little bit. Please, take a look at the in-line comments provided ;).

UPDATE 1: Ooops, accidentally pasted a line of code I used to test my way of 'thinking' (with the 'perl -e' command):

while(<STDIN>){chomp;last if m/done/i;push @a, $_;}print@a;
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Sorry about that, Anonymous Monk! ;)

_____________________

$"=q;grep;;$,=q"grep";for(`find . -name ".saves*~"`){s;$/;;;/(.*-(\d+)
+-.*)$/;
$_=["ps -e -o pid | "," $2 | "," -v "," "];`@$_`?{print"+ $1"}:{print"
+- $1"}&&`rm $1`;
print$\;}
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