Long Beach is a great little airport (emphasis on little)! If possible, try to fly into Long Beach. It’s about a 10 minute drive to the port. Jet Blue pretty much monopolizes most of the flights. Although you might find some options with Delta and America West. American no longer flys there. Steve’s description of LB airport resembling a bus terminal is a good one!
Flying into LAX will mean a longer trip to the port, mostly due to traffic (probably close to an hour). However, you will have a lot more options as far as air carriers. And the shuttle options are endless, try Prime Time or Super Shuttle for cheap fares. You can even reserve on line and get a discount.
The ports of Long Beach and Los Angeles are right next to each other. They are basically the same place. So either airport is an option. You may also want to consider Orange County airport (SNA) for other options. Traffic-wise, it might be close than LAX.
Have a great trip!