Actually, Dockweiler is the closest beach, just west of LAX. You'll see the planes flying over your head, and there's not much to do there off the beach, but there's a lot of sand and surf, so you could stop by for a quick set if you wanted to.
BTW, the spider isn't the old control tower. The old control tower is further to the east and it's now an airport office building. AFAIK, the spider, or really the "Theme Building," was always a restaurant/bar with an observation deck up top that is now closed.