In general, if I understand correctly, the representation theory of the braid groups is quite complicated, and there's no classification of the irreducibles. However, the braid groups form a sort of system of groups just as the symmetric groups do, and so one can ask about representation stability for coherent systems of representations of braid groups. Editing in response to Andy's comments below: it may be that "representation stability" doesn't mean anything because we can't decompose braid group representations in the same way we can decompose symmetric group representations. So a more basic question would be: is there any sensible way to talk about systems of braid group representations stabilizing that doesn't involve decomposition into irreducibles?

This is a very basic question, mostly just a reference request. 

Note that I am not asking about representation stability for the braid groups themselves.