I've had no trouble using my credit card in any port. Just make sure the store runs the card in US dollars, otherwise the credit card company can charge you a conversion fee.
American money is very welcome in ports. On our last trip I tried to get local coins as change in all the ports for my neices, but some places the shops only had US currency so it took some work to get local coins.
St Marrtin can be a little different since half of the island is on the Euro. It means that there your American dollars are easy to spend, but the conversion rate does not favor you.
Grand Cayman is another place you don't get as much bang for your American buck. It is a Banking and Jewelery center so don't expect the bargains you can get on the islands or in Mexico.
If you get away from the port areas, you may find your credit card less affective and Traveller's Checks harder to cash, but I have not experienced any problems spending money on any of our trips.