We drove to the pier in Manhatten in May for a cruise on the Spirit. We did not have any problem with parking and when we returned, we noticed that the parking lot was not completely full even though the ship was sold out. Please note that there are actually four (correct me if I am wrong) piers, that are all connected by the same ramp. You would normally park at the lot on top of the pier that your ship is docked at, however even if that lot were full, there are still the other ones available. The only difference is that you would have to walk further to get back to your pier.
Anyways, we were able to park immediately and right by the front of the pier where the elevators are located. You go to the first floor and the baggage men are right there to check your luggage.
When we returned, we breezed though customs and were in our car and out to the Holland Tunnel within 20 minutes of leaving the ship.
We were very pleased with the logistics considering this was our first time driving in NYC and have a brand new car. The ramp is well lit and guarded with bomb-sniffing dogs.
Sailed out of NY four times since 2003, and parked at the pier three of those, all without a problem. We arrive early, maybe 11:30 or so, and there is minimal traffic and plenty of parking available. Based on reading message boards, we never developed a back up plan due to the positive reviews. The only thing I try to do is check the NYC activity for that day to make sure we weren't running into a big event where traffic might be a problem.
As an aside, the only problem we have ever had was when we decided to take an NCL arranged bus to and from the pier because of our December departure. Upon debarkation, the bus waited about an hour for a passenger who never did show up, so parking at the pier is the only way to go for us. It is a little pricey, but you can't beat the convenience.