Closest airport would be Newark Liberty (Newark ,New Jersey) . There are airport hotels surrounding the airport and business hotels like Sheratons etc in the surrounding communities.
I don't think there is much of anything in the immediate area of the port in the way of hotels or even neighborhoods. As I recall the immediate area is industrial.
Pappa Bill is absolutely correct. Many of the hotels offer free aiport shuttles to and from the teminals. I know that the Holiday Inn and the Wyndham are among them. It would be worth checking to see if they also offer shuttles to the pier as it is not very far. Unfortunately, if you stay near the airport you really cannot venture out, it is not a great area. Closer to the pier, there is a Marriott Courtyard but there is nothing to see or do over there either with the exception of a hugh outlet mall. Good luck and have a great trip.
Newark AP (EWR) is the closest but La Guradia (LGA) and Kennedy (JFK) aren't that far -- maybe 30 miles tops, if that, which in NYC/NJ traffic can take 30 minutes or 2 hours. I'd check prices for flights at all 3 APs. There can be substantial differences. Then factor in the costs of transportation & time; don't forget the tolls.
The outlet malls are pretty good & you can get a decen, clean hotel room at a reasonable price ~~ around $85 per night, or cheaper if you hunt / bid. If it's your first time in the NY metro area you may want to consider splurging & staying in NYC itself but realize that is going to be about $200 per night for a tiny room.