The Baal quest is a quest just like every other quest in the game: if someone gets credit for completing the quest, all characters in the same act get credit. You get credit for completing the baal quest when you are watching him die.
That's what the "Baal 4 Quest" games are for: someone wants to move to the next difficulty level. I tend to be a nice guy, so I log in my mule/forgerunner first, to verify that the quest is available and to make sure I'm in the game in case it fills up. Then I ALT-TAB to the other game and log in my lvl 91, who'll help the game creator with the baal quest (or do it solo, I don't care what the other guy does, as long as he gets quest credit).
The game creator watches Baal die, and everyone --- most importantly my mule/forgerunner in town --- gets credit for it and advances to the next difficulty.
The same strategy works for the other act bosses as well; Duriel in particular. It's always "interesting" to see people come in, notice that we are already in the tomb, and leave a few minutes before we kill Duriel (or talk to Tyrael, for the hairsplitters) because they think they won't get the quest without getting staff and amulet and summoner quests first. They are just wasting their time.
It's best to think of it in terms of a "quest relay": someone who needs a quest needs to be around when "the thing" happens, but he will relay the quest credit to everyone in the act. I usually put my mule into the area when killing Diablo, though, since some people still manage to get killed (sorry guys, I can't completely shut down diablo...), and they hit ESC --- if Diablo dies while they are in town the quest gets lost. So if my mule is there then it doesn't matter what they do.