That is powdery mildew, a very common fungus affect many plants including dogwood.  The only thing to do right now is to make sure you clean up any and all leaves as they fall and put them in the garbage.  Good air circulation helps keep it from forming.  You could use a fungicide in the early spring if you wish.  There are some that are systemic and go into the plants circulatory system.  It will not kill your shrub.