I believe that each person has a personal exemption of $400, which can be combined among family members. So, you would pay no duty on the first $400 worth of merchandise. Over your exemption, I believe the duty is 10%, but they do not collect it if the amount is less than $10.
For example, if a husband and wife spent $600 on the ship and in Nassau, their combined $800 exemption would mean that they do not have to pay duty. If they had spent $1000, they would exceed their exemption by $200, so they'd pay 10%, or $20.
They personal exemption may be $600 for the Bahamas, as it is in the Caribbean, but I've heard conflicting reports about this. I also got different answers to the same questions onboard the ship, leading me to report to customs on the last morning of the trip, only to be told that we didn't owe anything, so we didn't have to.
There are other guidelines for alcohol and tobacco, but I'm not sure of them. Please note that I do not work for the US Customs service, so don't take what I say as fact. Check out the US Customs website (
link).