I highly recommend what body2body posted. I use the Flyaway if no one can take/pick me up from the airport, it is clean, affordable, efficient, fast and reliable. They leave Downtown LA (Union Stn) every 10min if I'm not mistaken.
Pick-up from LAX though could take a while, so better check their schedule. Once, I waited for 45min for the right bus to take me from LAX to Downtown, and that was just after flying 18 hours from Southeast Asia. Still though, a great way to get in and out of the airport.