No. In the Caribbean, you will not have to go through any immigration or customes. You will need your ship card and ID to re-ente rthe port area when going back to the ship, but that's all.

Returning to the US (or going to a US port after visiting a foreign port) you will have to clear customs and immigration. However, when we returned from a recent cruise to Tampa, the immigration and customs was extremely cursory. It consisted solely of turning in the customs form.