You'd more than likely need to sign up for an international plan so you won't be roaming. Make a list of the areas you will be hitting and make sure an international plan would even help in those areas. You will only get service while docked or near a tower though.
Use it as little as possible on the ship. Cruise ships charge their own rates and it's usually about $2.50/minute; I don't know about data. If you use it once you're off the ship at a port of call, the rates are generally lower, but find out the countries/ports you will be at, and then check on the website here or call customer service to get rates and value plans. Cruise ships usually turn off their tower as you come in to port and turn it back on once the ship is moving away from port. Remember that all text messages received (whether opened or not) are billed, and if you don't disable data background data will run and cause data roaming charges.
Saving money and using your phone on a cruise ship do not belong in the same sentence. Verizon's international plans do not include ship based use. You are better waiting until you are in port and using the local cellular provider with Verizon's international plan.
That was 4 months ago, I assume they already went on the cruise. I am commenting to clear up since my last post, Verizon does now have cruise ship plan one can use. Check out the International Trip Planner link posted above on Snn5's post