Finishing up my Babylon game. I've got 18 DV points and we just hit the modern age, so for the settings I play in, that ain't bad (huge map, epic pace, continents, immortal, no special rules). Thing is, while it's going well, it's not going as I expected.
As others here are saying, Babylon isn't really a science civ, exactly. My best guess was that they were more a diplomacy civ than anything else, based on the bonus envoys. The eureka thing is a fun mechanism, but it seems like more of a malus than a bonus, frankly: getting all those eurekas requires diversity, which is not generally how you win a Civ game. If you're a science civ, you can ignore faith and limp along with minimal culture, but as Babylon, falling behind in any one area limits your ability to get the eurekas you NEED in order to keep up. And I do mean keep up - I was NEVER ahead of the game, scientifically. At best, I was in the middle tier. The AI grabbed Great Library before I could and on Immortal, will vastly out-science most human players anyways. The eureka mechanism, therefore, seemed to be best viewed as a "rubber band" effect rather than an advantage.
That said, Babylon is actually really fun and highly competitive as a diplomatic civ. Canada relies on its no-surprise-wars trick to let it survive the early eras, but Babylon's UU accomplishes the same thing, while getting the player an early Golden Age and facilitating both scouting and barb-hunting. It's a strong defensive unit, and it helps you meet city-states early, meet other players, grab goody huts, and boost your military score as a deterrent (helped me get on Egypt's good side, too). Trade routes were also really easy to start, with a market and a lighthouse up early. All this resulted in being able to rack up DV points even after I dropped the ball a few times on aid requests (I prefer to save up and drop a lump sum the turn before it ends, on the theory that steady gifts will trigger the AI to try to keep topping the player; the problem is loading a save and then forgetting about the contest). Finally, the diplomatic approach helped my science by driving up my alliances quickly.
In sum: fun civ, strong early game, competitive in diplomacy, don't expect to out-science Korea. Theme song is a B-.