(26 April 2016: Updated to give a fuller answer.)
I'm fairly confident this question is still open.
As Ian Agol points out in comments, the 3-strand braid group is, by a happy accident, also the fundamental group of the trefoil-knot complement, which has been known to be conjugacy separable for a long time. (All compact, orientable 3-manifold groups are now known to be conjugacy separable.)
As far as I know, nothing has been published on the conjugacy separability of $B_n$ for $n\geq 4$. Quite a lot of recent effort has been expended trying to work out whether or not braid groups are virtually compact special. If that were true, then conjugacy separability would follow (at least for a finite-index subgroup) by Minasyan's theorem. Unfortunately, the wonderful recent work of Haettel seems to indicate that braid groups are probably not virtually compact special.
One can certainly imagine attempting a proof by induction, using the Birman exact sequence. But there are quite a few technical difficulties to overcome, so making an argument like that work would be a great achievement.