Sweep the snow out before it turns to slush whenever possible. A push broom is very effective for this.
Mop up any watery slush that couldn't be removed. The mop can be kept in a nearby bucket for quick access and frequent use.
Place a fan inside the garage. This will help dry up moisture. A fan will remove moisture even when water is frozen.
Designate an area for snow-covered shoes. When you walk into the garage you can leave your shoes at the entrance, or remove most of the snow from them. A large rubber mat works well to remove the snow and is easy to clean afterward.