When ever my young children cut themselves all was fine until they saw the sight of blood. Then the floodgates would open and they would cry all the more.
Well here is a simple idea. Always have a small red towel near the first aid kit. When you are bathing the wound use the red towel. If you are worried about it being sterile dampen the towel, pop it in a plastic bag and put it in the microwave for a few seconds. Make sure it cools down.