foreach my $name ( keys %birthdays) {
    if (exists $birthdays{$name} ) {
        ...
    }
}

If you're iterating over the keys of a hash (as in the quoted code above), each key must exist in the hash! One way to do something like what you want:

c:\@Work\Perl\monks>perl -wMstrict -e
"my %birthdays = (
   Fred   => 'June 1',
   Ray    => 'May 31',
   Glenn  => 'May 27',
   Bob    => 'August 28',
   Jim    => '',
   Paul   => undef,
   );
 ;;
 my $fmt = '  %-10s %s';
 printf qq{$fmt \n}, 'Name', 'Birthday';
 print '-' x 25, qq{\n};
 ;;
 B_DAY:
 foreach my $name (keys %birthdays) {
   next B_DAY
     unless defined $birthdays{$name}
        and length  $birthdays{$name}
          ;
   printf qq{$fmt \n}, $name, $birthdays{$name};
   }
"
  Name       Birthday
-------------------------
  Ray        May 31
  Bob        August 28
  Glenn      May 27
  Fred       June 1
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