Harrison Cave is a nice and well run cave, but it is small in comparison to some of the caves you'll find in the US. If you've seen other caves, this won't add much to your "cave experience". However there are some other nice attractions close to the Cave, e.g. Welcchman's Gully. Another data point is that the Cave gets very crowded in the heart of tourist season, with waits to enter via tram as long as 2hrs, but I've heard the average is more like 1 hour.
I love to snorkel, so my choice would be to go snorkeling. I've not found a really good location on Barbados to snorkel. Where are you considering going? Folkestone Underwater Park?
IMHO, I disagree. I thought for a Caribbean island, Harrison's was quite unique with its electronic tram. I have been to Bermuda, Skyline Drive, Polar Caves and Harrison's which I find were the best. Now you can go via Taxi tour right outside the port terminal building for much less than the ship. Get off early and go straight to the caves before the ship tours even arrive. Also, the Orchid World or Floral gardens are beautiful. Its your choice.
NanaTravel, I agree with you. It is an excellent Caribbean Cave. However compared to caves in the mainland USA, it is a minor cave. Like most questions, this initial one posed can't be answered without knowing the originators interests and history.
I also agree that if you want to do Harrison's, take a taxi and beat the ship excursions there to avoid the crowds. If you hit it during the peak time (i.e. when the ship excursions arrive), there probably will be a wait.
Of course, why not do both, see Harrisons and snorkel. Beat the crowds by going early in the morning, then snorkel in the afternoon :-)