The Port of Los Angeles is one of my all-time favorite places to paddle. The immense scale and complexity of the shipping operations are amazing and, frankly, a bit overwhelming at times. The water is always calm, with occasional wakes from passing ships, and often mildly windy. It's an industrial area, so it doesn't smell great, but you get used to it. No much wildlife, but there are seals, sea lions and lots of birds.
There are three places you can put in: Banning's Landing Community Center, Berth 84 (next to the Maritime Museum), and Cabrillo Beach. Cabrillo is okay as a strictly recreational area (which gets very crowded), but if you want to see the big ships and the inner workings of the Port, you'll want to put in at Banning's or Berth 84.
Banning's is by far the quieter put in. However, it is much farther away from the Battleship Iowa. It's great for watching the container ship operations and for paddling deep into the bowels of the Port. It's a very industrial, not-at-all pretty place to paddle, but I find it endlessly fascinating. One of the most surprising things I have seen is the large number of abandoned, decrepit boats and yachts in the various marinas. FYI, once you paddle past Schuyler F. Heim Bridge, you are officially in the Port of Long Beach.
Berth 84 has more to see by way of the "public" part of the Port, but it is also busier. It's safe, but you'll need to be watchful and mindful of the boat traffic.
The Battleship Iowa, cruise ship terminals, and the Vincent Thomas Bridge are a short paddle inland (northward). If you take a longer paddle south, towards the ocean, you will pass the San Pedro Fish Market and Ports O' Call Village. This area is where you will most likely meet some seals and sea lions, who like to pop their heads up to investigate your kayak.
Keep going south to the Port entrance and you'll see the Los Angeles Pilot Service on the west side. On the east side you'll see the Coast Guard base and and federal prison on Terminal Island. If you paddle further west, you'll see the S. S. Lane Victory, a WWII-era merchant marine cargo ship.
I highly recommend visiting the Lane Victory, the Iowa, and the maritime museum. In fact, there's a lot to see and do at the Port. If you get to one of the landings at daybreak, you could easily spend 3-4 hours paddling, followed by lunch and an afternoon visiting the museums.
It surprises me that I have never seen another kayaker at the Port, even though it is, in fact, perfectly legal (I've checked with the Port Authority). There is, of course, the safety issue, as there is with all kayaking. If you're an intermediate or advanced kayaker, you should certainly give it a try. Just be safe, be observant, and use common sense!
Be sure to read the links below for more info on the rules, regulations, etc.