If you still have the Beige G3 or can borrow another Mac, hook the two together and see if you can copy a folder full of small files. If they all make it, then the ethernet is most likely okay.
The G4 does use different ethernet circuitry than the G3. Does your cable modem have its own built-in dial-up or does it require the G4 to dial-up as well?
drewman