There are three distilleries in Barbados - Mount Gay, The West India Rum Distillery and Foursquare.
Mount Gay doesn't encourage visits to its distillery, but you can visit the "factory" just north of Bridgetown where they perform all the other stages of the production process such as refining, ageing, blending and bottling.
The West India Distillery is where they make Cockspur. I'm not sure if you can visit or not, but someone else will know. This used to be the Malibu Visitor Centre, and if you can visit it's very close to the Mount Gay factory.
Foursquare can definitely be visited. It's situated in the east of the island and it's where they make Doorly's, Old Brigand, ESA Field and Foursquare Spiced. Also, Pickwick Cricket Club's home ground is Foursquare Park, right next to the distillery.
Have you considered Grenada? It also has three distilleries - Clarke's Court, Westerhall and River Antoine.
When I was last there, in 2006, Clarke's Court weren't doing any tours, but I believe that that has now changed. They make about a dozen different rums, and you can sample all of them.
Westerhall don't distill their own rum any longer, but they import rum in bulk and age, blend and bottle it. Their Westerhall Plantation Rum is one of my favourites and is probably the best rum you'll find in Grenada and one of the best in the entire Caribbean.
At the opposite end of the island is River Antoine where they make "Rivers" in two different strengths: 69% alcohol and 75% alcohol. Only the first of the two is allowed on board an aircraft. Both of them are unaged, raw, white rums straight from the still, so don't expect too much in the way of quality. However, the distillery is well worth a visit, even if you don't sample the rum, as they have been making rum in exactly the same way for well over 200 years.
The Grenadian rums tend to be very difficult to find outside of Grenada, whereas most of the Barbadian rums are exported to many parts of the world.