What the others said :)

Separate the HTML output from the logic: this code has the main effect of returning a userid, and the side effect of outputing the HTML header. At some point, you won't want to do the two things together.

Why not return undef, or '' as the not logged in value? I hope those will never match a valid userid, and also allows you to write:

my $userid=getuserid or castigate_user;
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Use if as a statement modifier, rather than as a test in it's own right, then the code becomes more linear, and easier to read:

sub get_userid {
  my $member_dir = "../../Members/";
  my $member_file ="memberslist.cgi";

  return undef unless my %cookies = fetch CGI::Cookie;
  return undef unless $cookies{'UserName'};
  return undef unless $cookies{'Password'};

  my $id = $cookies{'UserName'}->value;
  my $password =  $cookies{'Password'}->value;

  my $exact_file;
  $file = "$member_dir/$member_file";
  open (FH, "< $file") or die "Could not open $file: $!";
  while (<FH>){
    chomp;
    my ($login, $junk_file) = split(/|!!|/);
    if ($login eq $id) {
      $exact_file=$junk_file;
      last;
    }
  }
  close FH;
  return undef unless $exact_file;

  $file ="$member_dir/$exact_file.cgi";
  open (FH, "< $file") or die "Could not open $file";
  my @user_list=(<FH>, <FH>);
  close FH;

  return undef unless $user_list[0] eq $id and
         $user_list[1] eq $password;

  return $id;
}
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Finally, do you really need to have an arbitarilly named password file for every single user? Why not just store the password directly in the members.cgi file?