There is no security advantage to disabling SSID broadcast; it's less security than hiding the key to your house under the door mat.
Further, by design most wireless clients look for the SSID when searching for a network. The standard does not address the exact mechanism for a client to find and connect to an access point that is not broadcasting an SSID, so different clients behave differently in this situation. Typically an AirPort card does not "see" an access point/router/base station that is not broadcasting, particularly when waking up. (There also seems to be some problem with computers running Tiger connecting after sleep, but that's a separate issue.)
I recommend that you enable SSID broadcast, change the SSID to something very different from the default "linksys," and of course change the admin password of the box to a secure password-certainly NOT the default "admin." If you choose something like "ThisIsMine" as an SSID be warned that not all access points handle mixed case the same way, so this might not show up as different from "THISISMINE" or "thisismine." Further, a good, unique SSID shows that you're paying some attention to security; the bad guys LOVE to find networks with default SSIDs, and they attack them more often because they're easier targets.
A final word: spend a few bucks and buy a more modern wireless router that supports WPA. WEP is "teh crap" in terms of security-not quite as bad as hiding the key under the doormat, but more like locking your door with a good-quality rope tied in a large knot. Yes, it is THAT insecure.