Lets explain. You live in Surrey, if you take a Surrey registered taxi into Kent, then he is not allowed to pick up a passenger in Kent so he has to return with an empty cab. Likewise in Bayonne, this is in New Jsersey, so if he drives you across the state border into New York, he is not allowed to pick up in New York and has to return empty. Therefore you will be charged a large premium for his 'loss of trade'. A cab from Bayonne to Manhattan will cost you around $110. However as they do not have to do the journey I actually had one taxi want $200 each way and another that wanted to do a tour as well as the ride for $450 on way! It should be cheaper to organise a private limo transfer - try http://www.carmellimo.com/
Or you could do what we always do which is to catch a NJ cab to Journal Square where you pick up the PATH train direct to 34th Street. PATH costs $1.75pp and the cab to the station should cost you around $35.
If you don't do the PATH train option I would organise (over the net) taxi/van/limo as the queues can be horrendous. Good Luck Wilba.
Book the tour from the ship. Going independent is not always the best way.
What tour? As the OP doesn't mention what ship he is on, we are unaware that there are tours. [a] RCI do not offer tours of Manhattan after their cruises for example and [b] I get the impression that the OP is going to be staying in Manhatten post cruise so will need to make his own way there.
Unfortunately Cape Liberty, now there's a misnomer, in Bayonne is the pits. Do as the others suggest, just getting to a hotel in or near the airport can cost $70 let alone trying to cross state lines into NY city. The Russians have bought the land around this dock but have not done anything with it to date, one day maybe but in the meantime it is awful (that's awful not awesome) and there are so many things wrong with this area so please do take the advice given. Ida.