1 check with the cruise line. My guess is probably YES because at some point you will step foot on foreign land and that is just how it goes. You will likely need to go through customs once you embark and disembark in the foreign county.
Now if the ship was out to sea (maritime law) and you went from US Mainland through other countries waters on the way to e.g., HAWAII, then NO. Also, should you get separated from the ship you would not be able to easily return to the US without a passport. It's easy to get one and they last about 10 years so I say go for it but it's getting close so I would go ASAP. Also, it is important to let them know that you have a travel date- they can rush it. There is a box you check.