Stay where ever you can get a killer deal. There are soooo many hotels down here, try to get something within walking distance to the beach. If you don't want to stay down in Miami, then Hollywood might be an good alternative.
Before the game? TAILGATE of course!! There are so many things to do down here it's hard to suggest anything without knowing your hobbies/interests. Renting a boat for a half day would be fun, most people don't get to see Miami from the water, its looks totally different from that perspective and tons of restaurants and shops are accessible from the water with docks.
Fly in wherever you get the best deal, it's a wash. FTL is less congested and generally easy in/out. MIA is usually a traffic nightmare, but at least you'll get dropped off in a spot where you can see the very cool skyline of the Magic City and really close to South Beach if that's your thing.
A couple good alternatives to South Beach is Hollywood beach Broadwalk, and Las Olas beach in FTL. There are water taxi's in FTL that travel in a big loop which you can get a day pass and check out all the stops.
