I drove the highway in November with a Bolivian client who had never seen the Rockies before. Snowed the whole damn day, and couldn't show him the most beautiful view from any road in Canada. Still got to Calgary in a day from Edmonton using the Jasper-Banff Highway. The next day it was clear blue sky. Don't worry about driving the road by yourself. It's a public well-maintained winter road, not a goat trail. Be prepared, as someone said though. Take warm clothes and some power bars no matter how nice the weather seems when you leave Edmonton. It's most unlikely you'd ever have to get out of the car if you didn't want to do so.