• taking a list and returning it spliced together as a string, only to split it again (why not return two array refs instead?)
• doing a split just to find out if there's a space character in a filename (why not just use m// ?)
• chomping filenames from readdir (where would the newlines have come from?)
• using @dir == "" rather than just @dir to detect whether an array is empty
• what does the code do when you have a file or directory name with a '+' or '%' character in it?
• re-opening your database for each file/directory

What's wrong with something like:

use File::Find;
use Cwd;
my $homedir = getcwd();

open DATABASE,  ">>$homedir/listofiles" or die "can't open listofiles:
+ $!\n";
open DATABASE2, ">>$homedir/listodirs" or die "can't open listodirs: $
+!\n";

File::Find::find( sub {
  return if /^\.\.?$/;
  return unless /\s/;
  if ( -d $_ ) { print DATABASE2 "$File::Find::name\n"; }
  else { print DATABASE "$File::Find::name\n"; }
  ( my $newName = $File::Find::name ) =~ s/\s+/_/g;
  print STDERR "renaming $File::Find::name to $newName\n";
  # rename($File::Find::name, $newName) or die "can't rename : $!\n";
 },
 $homedir
);

close DATABASE;
close DATABASE2;
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