Official parking at the port is $19 per day. This is a universal rate and there are no discount options. You can potentially park for free if you stay overnight at a nearby hotel and take the shuttle to the port in the morning, thus leaving your vehicle parked at the hotel for the duration of your cruise. Be sure to ask the hotel about their policy on park-and-go, however, to ensure they do not have a daily parking fee, or if they do, that you're OK with that rate before booking your overnight stay.
For more info, see our feature Bayonne Port FAQ