I second Long Beach, I've done the Westin there twice, and took a public bus to San Pedro for $1.25.  Then walked 15 minutes to the port.  Easy Peasy.   The whole LAX arrival / ground transport process is a nightmare followed by a $60 uber ride.  blah.  You can technically do it on public transport from LAX, but its time-consuming and involves transfers.  You can book a car service from LAX but its alot more expensive than the Uber.  Be warned that Ubers aren't super prevalent in San Pedro