Most Caribbean countries are now open to cruise lines at this time. One exception is the Cayman Islands.
If the country/port is open to cruise ships the current covid-19 restrictions will vary by country. Travel Weekly has the most up-to-date information in an easy to use global map format, see link below...
This interactive map provides detailed info on Covid-19-era travel restrictions, quarantine requirements, testing rules and documentation required for travel. Powered by Sherpa.
www.travelweekly.com
Once you comply with the covid-19 entry requirements there shouldn't be any restrictions for independent sight seeing other than the likelihood you may run into mask wearing requirements in some buisnesses, taxis, etc.