Once you are strong enough the way to take down slave trade is:
1. Find out if the slaves have a self destruct device if the master dies
1a. If they do than buy all the slaves
2. Kill the slave merchant and leg it
Killing them is the only way to get rid of the problem because buying slaves would just encourage them to capture more to sell, but if everyone that sells slaves ends up dead than no one would want to tame the risk anymore.