The best place to launch a kayak south of the Ben Franklin Bridge is Gloucester City Marina. The marina website doesn’t mention about kayaks but after checking it out for myself I found it to be quite welcoming, with plenty of parking, and access after hours. It’s a bit south of the bridge but the best place to do it.
If you are looking for something closer there are some other options but much less desirable. You might be able to put in by the Michael J Doyle Fishing Pier by crossing through the trees lining the north side of Jackson St to access the beach. Parking is limited and the beach might be on private property so I don’t recommend this option. I hope to save you the hassle in case you look at the map and have similar thoughts.
The public spaces in Camden, NJ near Wiggins Park Marina are all fenced off so unless you have a slip at the Marina and can access the dock or are willing to jump over the fence there aren’t many options available.