Not sure from the title of your post if you're looking for live halibut, or just live fish. I'm assuming its live fish, because halibut is pretty big to be kept in a tank. Foodiemahoodie's answer is excellent - what I would recommend. If you're willing to drive, and want very fresh fish that doesn't have to be live, Pierson's Port down in Newport is very, very good.
Some of the Korean markets in Koreatown also have live fish - actually, most of them do. The market in Koreatown Galleria and also the one in Koreatown Plaza do if I recall correctly. I recently saw a good selection at the big new market in one of the new malls on 6th. It might have been the one at 6th and Western or it might have been the one a bit east of there past Normandie on the south side - can't recall the name. Among the live fish I saw there were fish they called halibut but they looked more like flounder to me - not that I know the difference other than that I think of halibut as being quite a bit bigger.
I was at Ranch 99 in what used to be called San Gabriel Square - Del Mar & Valley, in San Gabriel, and they had a better even than usual assortment of live fish and shellfish in their tanks.
There is a fantastic Asian live fish market in Rosemead called Seafood Paradise Fish Market, 8955 E Garvey Ave, Rosemead. It's like visiting an aquarium where you can take the exhibits home and eat them.
99 Ranch Market in Van Nuys has live fish. I know Shun Phat markets do - the one in El Monte has the most and is worth a trip if you're serious about such stuff. Big jazucci-sized tanks with whirling water and live fish. And a variety of crabs and lobster, etc. Wish there was one on the westside.
Downtown L.A. also has some - I'm pretty sure LAX-C does (I've bought live lobster there a few times, but that's about all I remember).