$$ on board

Is it ok to use my debit card on board (which has a master card logo) or should I bring cash or travelers cheques and put it on my account?

You can register your credit card at embarkation, and it is automatically charged to your account at the end of the cruise. You can check on your spending during the trip.

If you do not register your card, then you can pay with travelers checks or cash.
TWS in NYC

Thank you for answering my question.
My next question is should I bring traveler's cheques for spending on board (drinks, stores, etc.) or can I just charge everything to my room?

[Cruiserking]

It's always safer to bring travellers checks, you can store them in your stateroom safe and easily cash them in at the purser's desk.

Cruiserking

[Marc Thompson]

It is a cashless environment on board. You sign for everything and it is charged to your credit card. You cannot pay with money or travelers checks for purchases. (Maybe at the end you can, but not on an individual purchase basis.)

[lexxity]

you register everything on the keycard you are given upon embarkation. If you are paying cash, when you get onboard, you need to go staight to the pursers desk and put money over the desk against your account. We have done this and you just need to make sure that you keep up to date.