You might try St. Barth, which gets a few ships, but they are all smaller ships and don't really disrupt the flow of island life. Also Anguilla.

In the BVI, Tortola is getting some cruise ship, but not Virgin Gorda, and never Anegada.

St. John in the USVI doesn't get cruise ships, but it does get a big influx of daytrippers every day who land on nearby St. Thomas.

The Grenadines are very quiet and tranquil. You could try Palm Island or Petit St. Vincent, both private-island resorts.

I can think of several destinations that don't have a cruise ship port, but even some of those have cruise ship visitors from other islands.

Anguilla and St. Barth don't have cruise ship ports, but visitors can come from St. Maarten. The impact is minimal. Similarly, Nevis doesn't have a cruise ship port, but visitors make trips from St. Kitts. Likewise, since there is not much for them to do, the impact is minimal. The Out Islands of the Bahamas (Abacos, Exumas, Andros, Cat, Long, Eleuthera, etc.) have no cruise ship stops except for a handful of "private" islands used by the ships and off the land-based visitors' typical path. However, there are excursions from Nassau to the northern Exumas, for example.

Of course, there are plenty of other considerations in choosing an island destination. Time of year you are traveling, budget, where you are traveling from, what do you consider "relaxing," etc. Some destinations might meet your requirements notwithstanding cruise ship traffic because of how well that sort of tourism is managed/absorbed, and how well the destination otherwise fulfills your wishes.

Yes, I forgot to mention Nevis. That's a great choice, even if you can't afford the Four Seasons.

Provo in Turks and Caicos. Theres a cruise stop at Grand Turk, but that's a flight away, so no chance of "day Trippers" either.

Anguilla, while it might get day trippers, actually stays pretty quiet.

Out Islands in Bahamas are laid back and relaxing.

Barbuda, part of Antigua. (no day trippers from Antigua either)

Anegada in BVIs.

Grenadines

Culebra (part of Puerto Rico but too far for day trips)

So tell us what else you're looking for, what's your budget, what time of year. To help narrow down.