If you want a view, you've come to the right place!  Banff is surrounded by mountains, and a window seat in most restaurants will give you something great to look at. But, as always, the canny traveller looks for the best of the best.  For dining with a view, the best of the best are the following:

Downtown 

All of these demand good weather ...

The roof terrace at the Rose & Crown (202 Banff Avenue).  A pub, with British-pub-inspired food, but in a beautiful location.

The roof terrace at the Elk and Oarsman (119 Banff Avenue).  This is a pub/restaurant, with a casual menu (burgers, steak sandwiches, etc.) and a good choice of beers.  The location is spectacular. They have the highest dining terrace in the downtown, and it's wonderful on a warm summer evening.  Menu can be found online.

The tiny (six-table) balcony at the Sukiyaki House (211 Banff Avenue).  Great sushi and other Japanese specialties, pleasant service.

The new addition to downtown - the Bison Courtyard - has a very pleasant deck with great views.

 Around Town

The Juniper Bistro (at the Juniper Hotel, off the Mount Norquay road).  The room is beautiful, and usually quiet, and the view takes in the whole sweep of the Bow Valley from the Fairholme Range to the shoulder of Bourgeau, including Rundle, Sulphur, and a distant view of the Banff Springs Hotel.  Great food (interesting, quirky, tasty, original), reasonable prices,sometimes the service staff is a bit inexperienced.

The top of the Sulphur Mountain Gondola:  yes, there is a restaurant up there!  In fact, three - the Summit Cafe, Altitude 1583 Patio (summer only, weather dependent)  and the Panorama Restaurant.  The restaurant atmosphere is pretty basic, the food is boring and pricey for what you get, the staff are rushed.  But this place has THE BEST VIEW.

The Fairmont Banff Springs Hotel has several restaurants with excellent views. The dinner buffet at the Bow Valley Grill is not cheap, but the food and variety are fabulous, the ambience is elegantly casual, and the restaurant overlooks what William Cornelius Van Horne, one of the original builders of the Canadian Pacific Railway, called "the million-dollar view".  And he was 1.  talking in the 1880s, when a million dollars was a LOT of money and 2.  a man who'd seen a LOT of views. The same view is available with a much more intimate ambiance from the Rundle Lounge, a small area surrounded by wall-to-ceiling windows. The Rundle Lounge offers high tea at certain times, and at other times drinks and an elegant light meal/appetizer menu is available. It's well worth the price of admission (a drink or two), although it can get hot here on sunny summer days. The Waldhaus Restaurant has very nice views of the Bow Valley and Bow River from some of their tables. The Waldhaus Pub has a patio with the same stunning views; the indoor portion of the pub has no windows. Both of these are in the same small building, down the hill from the main Fairmont Banff Springs Hotel or up the very long flight of stairs in the mountainside next to Bow Falls (look for the large sign by the Bow Falls parking lot).

Three Ravens is a formal restaurant in the Banff Centre, on top of the Sally Borden Building with great mountain view. The nearby Vistas dining room is a market-style restaurant with the same great views in a more casual setting.  

Well Outside of Town

Storm Mountain Lodge is located in Banff National Park in the Vermilion Pass on the Banff-Windermere Highway (Hwy 93 S). It is just 25 minutes outside of Banff and Lake Louise townsites. Storm Mountain Lodge, founded in 1922 by the Canadian Pacific Railway, offers stunning views and a rustic charm. The dining room is situated along the wraparound windows in the lodge, not far from the lounge, which is warmed by a massive stove fireplace. In the evening, guests experience delicious Canadian cuisine prepared with a dedication to organic ingredients and naturally raised Alberta meat and fish. Entrées include buffalo, wild salmon, rack of lamb and free-range chicken. All of the breads, desserts and breakfast pastries are baked on site from scratch. The dining room has an extensive wine list drawn exclusively from carefully selected Canadian vintages.