Semi-unofficial place to quickly put in your kayak, to tour the Port and have a close look at the Battleship Iowa and other large ships. there are also massive suspension bridges and cranes that are mind-boggling huge.
The best time to go is as early in the morning as possible, as boat traffic increases a lot by 9-10 AM. Obey the posted rules, practice common sense and situational awareness, and be safe!
According to the Port Police officer I spoke to (they and the coast Guard have stations here), the only "official" place to put in is at Cabrillo Beach. However, it is well understood that there is a lot of recreational boat traffic there, so paddling your kayak into the Port proper can be a little dodgy (literally and figuratively). He told me that as long as I didn't spend a lot of time hanging out on the landing, and obeyed the 100ft distance-from-ships rule, there would be no problem. In other words: get in fast, stay out of the way, and be safe.