When you're staying in Basseterre and you don't want to take a taxi all the way to Frigate Bay to go out: no worries! In downtown Basseterre there are enough small bars to go to. These bars are scattered all over the town and are easy to find: just walk around and follow the music.
The typical Kittian bar plays load Caribbean music, and has a very mixed audience: tourists, foreigners based on the island and locals. The typical drink is the local Carib beer, but a nice Caribbean rum is always a good option as well.
It's nice to go to bar like this with some friends, but if you're there alone you'll find people to talk with easily as well. I've been to a small bar close to the central park of Basseterre where there was a drinking- and an eating spot next to eachother. So after a couple of beers you don't even need to go somewhere else for a tasty meal.