The answer in many cases is neither, and in other cases one or the other! I'll try and explain.
On many of the Caribbean islands it is possible and very easy to do your own thing for a huge saving over the ship prices. Many of the ports have taxis waiting and often a despathcher with whom you can negotiate a price. Try and join up with others on the ship for a real bargain! Just make sure you get back to the ship in plenty of time - but seasoned cruisers will tell you that it's not a problem.
There are some trips that are difficult if not impossible to organise on your own, such as diving, sea plane trips etc and for these you will have to book the ship's trips. If it's a trip you really do not want to miss, book well before you go, otherwise wait.