Here is a list of Dreamcast games still working online. I got this from the GameFAQs Dreamcast forums.

US and EU Games Online through official servers:

4x4 Evo (both full and demo versions)
Maximum Pool (requires using this game save: http://www.pauth.com/download/dc_max_saves/MAX2.VMI )
Sega Swirl (requires a fully operational POP mail account)
Starlancer
Phantasy Star Online (European Version)
Phantasy Star Online Version 2 (European Version)


JP Games Online through official servers:
Note: Japanese games cost a fee to play online.

Phantasy Star Online
Phantasy Star Online Version 2


Games online through private servers:

Quake 3: Arena (various privately hosted servers)
Phantasy Star Online (All Versions) ( http://pages.globetrotter.net/ctrep/ )
Phantasy Star Online Version 2 (All Versions) ( http://pages.globetrotter.net/ctrep/ )

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I've also come across this guide on how to connect a Dreamcast to a broadband Internet connection using its built in modem.

This method involves you connecting the DC to a PC with a modem and then sharing the PCs internet connection with the DC.

The full guide can be found at: http://www.consolevision.com/members/mterlouw/