There are a lot products out there to help lessen the amount of snowballs building up on your dog's coat and feet.
First keep your dogs feet trimmed, trimmed feet has less hair for the snowballs to stick too.
Slicker, coats that cover legs and much of the belly also help. Many brands of dog boots are also available to purchase. If you want to go without protective gear, mushers wax repels snow from sticking to your dogs feet. Even baby oil will help keep snowballs from occurring between the toes.
I haven't yet tried this but some people spray their dogs feet and legs with Pam creating a slick non stick surface.
What ever your course of action removing snowballs can be stressful and painful. So if you use a blow dryer please keep it on low heat to melt the snowballs. The quickest way is a warm bath and towel dry well.