Downtown LA isn't bad at all. If you venture out past the downtown area you could end up in Shady areas, and you'll see signs that you might want to turn around before you get yourself into danger, but you're perfectly safe downtown. The hotels downtown are right in the center of downtown. You'd be safe to walk around for shopping or dining. To go sight seeing you'd be taking the freeway to your destinations and you're totally safe there.
Public transportation down here is horrible. If you can use the trains to get where you want to go it will take forever. Your best bet is to rent a car. (by best bet, I mean you'll really regret it if you don't)
If you're really uncomfortable about staying downtown, you could stay near Universal Studios, but that area is kind of boring, IMO.
Carson, yeah, I wouldn't stay there. Dont stay right near LAX either, you can get 15 minutes away from there in Manhattan Beach/El Segundo/Hermosa Beach and be close to the beach in very safe areas yet close to everything. It really comes down to what your personal preference is to where you want to establish as your "home base" during your trip.