at first glance the two if-statement blocks differ only in the constants "red" and "blue".

You can factor them out in a common sub:

sub good_sub_name_here {
  my ($color, $other_color) = @_;
  if ($old_color eq $color) {
    if ($_ == 1) {
      alert(1);
    } elsif ($button[$_-1]->cget(-background) eq $color) {
      alert(2);
    } elsif ($_ >= 2 && $button[$_-1]->cget(-background) eq $other_col
+or && $button[$_-2]->c
+get(-background) ne "white") {
      alert(3);
    } elsif ($button[$_-1]->cget(-background) eq "white") {
      make_color ($color, $_-1);
      make_blank ($_);
      check_win();
    } else {
      make_color ($color, $_-2);
      make_blank ($_);
      check_win();
    }
  }
}

# and call it:
good_sub_name_here(qw(blue red))
good_sub_name_here(qw(red blue))
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If you declare the sub in the place where the if blocks are now, you shouldn't have any scoping issues