Do you have WPA2 encryption on the router? Old Computer implies that the wireless adapter is far too old to support WPA2 (which requires different hardware than WEP and WPA require). Windows XP SP1 lacks a great many networking, especially wireless, fixes and improvements.
So, at a minimum you need to get that beast updated to SP2 and make sure you have the latest driver for the wireless adapter that can support WPA2.
I expect that you then will still need to back off your router to WPA or else get a new adapter that supports WPA2 fo the old computer.