I agree with Wilba pay by credit card but at the bank on board ship you can cash your travellers cheques for cash to use in the casion or on shore. The bank will have dolars and Euro's.
The easiest way, & the most common method is credit card. The card details are taken prior to embarkation & you will receive a statement at the end of your cruise with your all of your spend detailed. Most lines do not accept Debit card & yes you can pay cash or travellers cheques as an alternative, but it does mean queuing up at the pursers desk on the last morning with hundreds of others with a multitude of reasons for being there to hand over your cash. Personally I can't think of as worse way of spending the last few hours on a cruise ship. Whatever you choose, Enjoy It... Wilba.
I've been on the Destiny twice, and did the Med last August...city sight-seeing was VERY hot. Wonderful ship and crew, you are in for a great time. Do take some Stugeron with you (from Chemist) JUST in case you feel queasy - the Destiny is a bit wobbly! They swipe your credit card when you arrive, then all expenses on board (for drinks, tours, etc.) will be debited to that card on your departure. Drinks and tours reasonable prices. This is the easiest way to settle your account. I'd forget travellers cheques.