Once you execute, you still have to fetch your result set. So the following should work better for you:

 my $usr = $dbh->prepare(q{SELECT user_id from user where user_id = ?}
+)
   or die("usr prepare error: $!");
 $usr->execute($user_name)
   or die("usr execute error: $!");
 while (@row = $usr->fetchrow_array ) {
   if ($user_name eq $row[0]) {
     # Blah
   }
 }
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You should also be checking the return values to ensure your statements are being handled properly.