If you are using an AudioPlayer object, as in the "LoadFile" example included with Minim, then you can repeatedly poll AudioPlayer.isPlaying() to see if the player is still playing. When it returns false, the clip is done.
For example, in that example, you could add these lines to the end of the draw method:
if (!player.isPlaying()) {
println("DONE");
}
Note that the code will repeatedly trigger after the sample is done. So you might want to use an additional (global) boolean to force it to only trigger once. Outside of draw() you would add this global:
boolean triggerOccurred = false;
and then you would use this code in draw:
if (!player.isPlaying() && ! triggerOccurred) {
println("DONE");
triggerOccurred = true;
}