I have sergers, but I usually use just a straight stitch with my regular machine to attach borders and bindings.
I have used a "long" stitch (to minimize flattening the edge too much) with the serger on the outside edges before attaching the binding.
Serging does add more thread to the seam allowance.
Try it on a few scraps and see if you like the result. I don't think it would be "wrong" - but I don't see any benefit to serging the borders or bindings on.