Are you talking Calgary in canada? if so, then calgary airport in Alberta province, and Banff in B.C. province , like two difference states. few hrs drives. most ppl will rent the cars to get there, or take the small local plane from local airport in calgary to get there.
other options, train or Great Coach, Great grey(long distance bus), pick up point downtown bus terminal, this way will be cheaper then airport shutter bus or by CP(Canadian Pacific) Rail.
by the way, “Its about 120km/75miles looking at googlemaps!”, may be is true, in fact, is not, most driving is hillside or falling rocks ahead, so you can’t drive very fast speed. but very nice drive view, they called it little Switizland
P.S check the bargain for car rental, often I paid $13-15 Canadian + tax + free milliage economy size few years back, just when you get to the airport, ask around the bargain price (they don’t usually list in the internet) and have your plan B in place.
Be aware, most bus are drop off at Spring Hotel, if you don’t stay in that hotel, you need to take another taxi from there to your desire hotel