Hmm,
Doom Legacy is the first source port of
Doom that I've played, and it's quite fun. Also TonyD who did the bots there (I think) is the original author behind
Dune Legacy.
However, I'm pretty sure
Doom can also be played online via the DOSBox IPX emulation. There's even a community that plays DOS games like that (although they focus on more obscure titles),
The DOS Multiplayer Archives.