It would be foolish to trust the Dementors after they sided with Voldemort, but somebody's got to guard Azkaban- was this ever adressed?
According to the Azkaban entry on Pottermore, Kingsley gets rid of the Dementors and replaces them with Auror guards:
Under Kingsley Shacklebolt, Azkaban was purged of Dementors. While it remains in use as a prison, the guards are now Aurors, who are regularly rotated from the mainland. There has been no breakout since this new system was introduced.
We can only speculate on where they went after this, since Azkaban seems to be the only place they've ever called home; in fact it's strongly implied that they were created on that island.