The closest airport to Jasper is YEG - Edmonton International. Your options for getting there are:
1/ Greyhound, and then the Skyshuttle service from the city to the airport
2/ VIA Rail - but the connections are not good; I recently looked at this myself, for a flight this coming summer ... VIA is scheduled to arrive in Edmonton at 11 pm, too late for the ~1 am red-eye flights, and if you have to stay over in a hotel, that adds to your cost. I would never travel on VIA with a tight connection either, just in case.
3/ the Sundog airporter shuttle, which wil pick you up from your accommodation in Jasper and take you right to YEG, for about $90, if I recall correctly. This is by far the easiest and most convenient option... my father uses this when he visits me. (See the link in Country_Wife's post.)
4/ car rental
5/ Ride share (facebook) - a bit of a long-shot and would be more uncertain and you would not be able to know until not quite last minute, but you could always use one of the other options as a back-up. Basically, there are two active facebook group pages where people either offering or seeking rides to/from Jasper can post. It is customary to pitch in toward gas cost. If you are a facebook user, you can use the search to find them. They are called "Jasper Ride Share" and "Jasper, AB Buy, Sell and Trade!"
Edited: 4:56 pm, May 01, 2013