Trying to tell you the spots in Raritan Bay would be like trying to name all the stars in the universe...there's just WAY too many locations.
Here's just a short list of the more known spots....
1. Keyport...there's a small pier that you can crab from that's pretty good.
2. Keansurg pier...nown more for the fishing than crabbing....you can catch a decent amount of crabs there. There is a small fee to get on it though.
3. Atlantic Highlands Municipal Marina...there's a small pier here that you can crab from.
4. Port Monmouth....a very small canal/inlet that you can crab from off the wall.
5. There are small docks all along the Shrewsbury/Navesink rivers that you might be able to access that you can crab off. Most of them are private, but there are a few that you can crab from....some are just abandoned, or near bridges, etc.
6. Boat Rentals.....you can rent a boat at the Atlantic Highlands Marina, or at a few small boat rentals in the Highlands, close to the bridge leading out to Sandy Hook. Also there is at least one boat rental that I know of in Redbank (Navesink River), upriver from Sandy Hook/Sea Bright. No matter where you rent a boat from, I would recommend going upriver, in either the Shrewsbury or the Navesink. It's usually far better upriver, than closer to the Bay.
7. There are probably hundreds of bridges that cross the rivers/creeks along the bay and up the larger rivers that you can crab from. As long as there aren't any "no crabbing" signs, and you feel safe with the cars pssing only a few feet away, they can be some of your most productive spots.
Good luck out there, and leave a few for me when I come back to NJ to visit my family