Possible application to backboard in shower
I would let that dry the best you can, heat or fan...once that is done, paint it with kilz or similar and, of course, wait for that to dry, then install backer board up the wall. The reason I would paint the framing is to inhibit any growth of spores you can't get to die completely, therefore sealing them in and causing them to die. Some people seal it, others don't. I know a few people who bleach it and be done with it. Some use a mold/mildew converter, and some paint it with regular paint.