Staying near LAX has it's benefits. You're close to the Pacific Coast Highway, 105 and 405 Freeways so you can go anywhere you need to go, but having said that, the same applies to anywhere you stay. There are some disadvantages staying near LAX. For example, if you want to head down into OC, it's just THAT much further and you have to deal with traffic on the 5, 57,10 and 405 freeways. It's an absolute pain in the ass. Trust me on this one. That and parking is NOT free at LAX hotels unless it's part of your package. There are no "good" restaurants within walking distance of LAX. One wrong turn and you're in Inglewood and it's not a place you want to be after dark. You either drive northbound or southbound on Sepulveda within El Segundo or Manhattan Beach for some food options or you take the PCH into Marina Del Rey/Venice which is at least a good 15 to 20 minutes depending on traffic.
Yeah, don't stay in Downtown LA. You could try some hotels near Little Tokyo or the Art's District so you have some food options at night, but even then, some places are an absolute dump and not safe to walk around at night. And if somebody in these areas says they like your shoes, you better be prepared to give them up.
What do you have planned? Maybe that would give us a better idea of where to suggest. (I spend a lot of time in LA so I know the area pretty well.)