This is what I use:

BEGIN {
  require 5.004;
  use CGI::Carp qw(fatalsToBrowser carpout);
  my $logfile = 'path/to/logfile';
  open LOG, ">>$logfile" or
    die "Couldn't append to $logfile: $!\n";
  carpout(\*LOG);
};
[download]

This appends to the logfile all the output sent to STDERR, both during compile-time and during run-time. E.g., if I have the line

warn "Logging...\n";
[download]

in my code, this will send the following to the logfile:

[Thu Apr  7 05:39:57 2005] testcgi: Logging...
[download]