Remember Colonel Flagg from M*A*S*H, who hid in trashcans looking for Commies? You don't exactly want to become him, but on the other hand, you don't want to ignore the possibility of getting zapped by a virus or your network being invaded by hackers. Make sure that your Internet connection is properly secured with a firewall and don't allow any Internet access that circumvents your security.
To counter virus threats, use network-aware antivirus software to ensure that every user on your network has up-to-date virus protection. And teach your users so they know how to avoid those virus threats that manage to sneak past your virus protection.