Vitamin E is a good, safe remedy for dry skin that can be applied externally or fed to the dog as a supplement. Adding healthy oils like fish oil or olive oil to the dog's diet can also help.
To apply vitamin E externally, massage the oil into the dog's skin. Make sure it gets on the skin and not just the fur. It can also be added to a bath if the dog likes to soak. Avoid bathing the dog too frequently and try not to use a blow dryer to dry the dog, as this can dry out the skin. Bathing the dog in oatmeal or a moisturizing shampoo can help sooth itchy skin and add moisture, however.
Olive oil, fish oil and coconut oil can all be added directly to a dog's food; most dogs enjoy the taste of these oils. Only use 2 to 3 teaspoons a day and introduce any oil gradually, because otherwise it can cause diarrhea.
Dry food can have a desiccating affect, as the body uses moisture to digest it, so supplementing these types of foods with probiotics to support digestive function can help some dogs. Switching to canned or moist foods can also help.