Multiplayer for Duke3D is still alive and kicking (but well, not as strong as for Doom).
Modern online dukers use multiplayer launchers (graphical frontends used for connecting to hosted games by others), most popular are:
DukeMatcher and
Meltdown
Old school classic gamers prefer to use original dos executable Duke3D emulated in DosBox, but most widespread nowadays are souce ports, which allows you to play Duke3D natively in modern operating systems.
From them the most popular source port for multiplayer is definitely
hDuke. It could be argued that is most network code stable
as yet. Some priase its authencity to original dos Duke3D feeling, but others criticise it for lack of any modern functions - no hardware accelaration - that means no OpenGL powered rendering (no support for HD resolutions or correct widescreen aspect ratios, no texture filtering, no support for additional 3D models and hi-res textures, no dynamic lighting, etc ... ), same map and code limits as old DOS Duke3D for user maps/mods, lack of any special modern features, ...
On other hand is Eduke32. It features almost everything that I listed missing from hDuke except for stable network code,
as yet. This is caused by effort of its developers to offer us brand new modern network system that sadly still isnt finished so it is in pretty broken state at this moment.
BUT, there is partial remedy for this,
special edited version of Eduke32 with old network code was released that is reasonable multiplayer stable, so you can try that in meantime while new network code gets finished.
So pick your choose according to your taste.