Try doing your connection and initial test to syslog in an eval block. Then check $@ for an error. This will allow you to continue operation if syslog is not down. You can also create a generic log subroutine that eval's every syslog operation.

sub log {
    eval {
      syslog($_[0], $_[1]);
    };
    if ($@) { print STDERR "Syslog error: $@ $!\n";
}
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BTW, it would be helpfull if you would post the OS and version, as well as the errors your program crashes with. Also the contents of $! if possible.