The port of LA is a joke, but don't worry, I have yet to see any port except maybe Vancouver that isn't a joke, at least from the outside. As for taxi fare, I don't know, about 5 years ago it ran in the neighborhood of $35.00 but am sure it is much higher now. I would suggest you see if you can get another couple to share with you. You might even meet someone at the hotel the night before, who knows. Anyway that would be the least expensive way and kinda nice to get to know someone before sailing.
Good luck and if you have any questions I am sure we would all be glad to answer. NMNita
Been there, done that.
I flew from Florida to LAX (6:00 a.m departure;arrived about 10:30 a.m.), took advanced reservation Super Shuttle to the pier (San Pedro), and checked in RAPIDO! I was onboard the STAR about 12:30 p.m., ate like a pig, and then took a well-earned nap. My cabinmate woke me up when he arrived about 2:30 p.m.
Net, net: I advise going straight to the ship. The pier is commercial and like any other you will dock at anyplace else. LAX has any transportation that you could ever want! (check the 'net). I promise you that you aren't the first persons to need to get from the airport to the ship!
enJOY the Mexican riveria! Blessings on your voyage! Go watch the cliff divers!