If you are in Kotor, Tivat, Budva, Herceg Novi or arriving by cruise ship, the Montenegro Blue Cave is the easiest and best-value option. It is a 40-70 minute speedboat ride from Kotor, it is usually part of a 3-hour route with Our Lady of the Rocks, the submarine tunnels and Mamula Island, and it is the only famous Blue Cave where swimming inside is the point.
Croatia's Blue Cave on Bisevo is spectacular but logistically heavier from Split, usually expensive as a full-day island tour, and swimming has been banned for years. Capri's Blue Grotto is historic and atmospheric, but the actual cave visit is often only a few minutes in a rowboat after a long queue.