Heh heh. You said "Banff" and "Away from Crowds" in the same sentence.. Fail.

Ok, it's actually not that bad but from someone who spends a lot of his time up north in Jasper, Banff often feels like a zoo to me

Having said that much of the actual backcountry isn't terribly crowded. (the day use trails will be, particularly in the lake louise and banff townsite areas) If you are backpacking in the park you do need to check with the park office and get permitted. You can either do it on a daily basis or buy a yearly wilderness pass (the break even point is about 5 days I believe - I don't look because I always buy one

Depending on how many days in a row you want to be out, you may want to think about the Skoki area (middle of banff park) or at the far northern end of banff, Nigel pass, then through to Jonas and Poboktan (Jonas and Poboktan are actually in Jasper park) - You may see that hike described as part of the "Brazeau Loop". The Mount Assiniboine area is also nice but you can expect a fair number of people there. (and helicopters)

If you're just looking for other stuff "Near Banff" there are
some other good possibilities in the area, Peter Lougheed Provincial park which is adjacent to banff has some nice stuff . An in and out to Turbine Canyon and back out would be a very nice trip you could do overnight or in a couple of days.

One other excellent possibility you may consider nearby is the Rockwall Trail in Kootenay National park in BC (also adjacent to banff). Rockwall is pretty spectacular and will have less people on it than anything in banff. it's about 5 days at an average pace.

For reference, here's a Nigel Pass hike:
http://bofh.ucs.ualberta.ca/beck/pictures/nigel07/

Here's Rockwall:
http://bofh.cns.ualberta.ca/beck/pictures/rockwall2007/