Take the bus from Edmonton to Calgary, 3 hours. Greyhound is downtown Edmonton, rather ghetto too
Brewsters I believe runs a bus from Calgary to Banff, guessing from the airport. Or just ride it out from Calgary to Banff (130 km or so), a great if you take Highway 1A. Or hitchhike, lots of tourists with RV's coming out of Calgary.
Another option is go from Edmonton Airport to Jasper on the Sundog Bus. They take bikes no problem, I've taken dozens of times (I live along the route). Once your in Jasper, just hitchhike or ride down the parkway to Banff
It's worth it trust me.