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You can use any of the choices you mentioned, but they prefer Credit Card as it makes life easier for everyone. Do you really want to stand in a queue the length of a football field on the last morning to settle your account? Not the best way to end a cruise. Put it on the card, pay it off in full at the end of month, job done!
You can go and make cash deposits [if you have a winning streak from the casino !]during the cruise, but as Wilba says, put it on your creditcard. No debit cards accepted, and I wouldn't bother with travellers cheques, as you have to convert these to cash first so it is still a hastle.
When you check in to board the ship at Miami the best answer is to show your credit card, which they swipe through the computer and return there and then.
After that the card they give you to open your cabin door also acts as a payment card for anything (drinks, spa, any shopping goods, shore excursions etc) that you "purchase" on the ship. You can chech your on-board account at any time via your cabin TV set. If you feel a mistake has occured, sort it out right away rather than leaving it till the last day. That way you can keep tabs on your spends without having to carry money around - except, of course, if going ashore.