Visit one of the museums listed on the Inside Banff : Museums  & Attractions page here at TripAdvisor.

Bathing in the outdoor pools at Banff Upper Hot Springs is often more enjoyable when it's raining or snowing.

Visit the Fairmont Banff Springs Hotel. This historic "Castle in the Rockies" is open to the public, so you can explore this amazing Gothic-inspired hotel on your own; ask for a copy of the hotel map, "Your Key to the Castle", at the front desk. In the summer (mid-May to mid-October), a guided tour is available for non-guests in conjunction with brunch in the Bow Valley Grill. Call (403) 762-2211 for details. Groups may arrange tours year-round by calling (403) 762-6827; there may be a charge for this service. A guided tour, however, gives you access to parts of the hotel which are ordinarily closed to the public, and your tour guide will tell you about the hotel's fascinating history. If you're in a hedonistic mood, check into a visit to the hotel's Willow Stream Spa (reservations recommended).The Banff Springs also has shopping in a variety of stores, ranging from CPR-related items to fine clothing to Canadian wines. There are also many places to eat, in a wide price range.

Five-pin bowling at Fairmont Banff Springs Hotel : The phone number for the Fairmont Banff Springs Bowling Center & Striker Lounge is (403) 762-6892.

Watch a movie at Cinema Lux Centre at 229 Bear Street.

There is a saying, "When the going gets tough, the tough go shopping."  If that is your approach, the many shops in the Town of Banff certainly can accommodate your love of shopping

Although Banff townsite's museums and shops could keep a family meaningfully occupied on a rainy day, the town is too small to hold your interest for days on end.  The best thing to do is to come equipped with clothing that will enable you to get out and about in any kind of weather.  Note that rainy days are also quite cool; there is no such thing as a "warm rain" in the Rockies.

The Lake Minnewanka boat cruise can be a good option on a rainy day, since the boats are enclosed and heated. (There is also an exterior deck).  Although the rainclouds may obscure the surrounding mountaintops, rain also means reduced human activity, so wildlife sightings can be improved.

A good outdoor activity on a rainy day is a visit to a canyon, because you can enjoy the beauty of a canyon even when visibility is poor.  The canyons in Banff National Park are Stewart Canyon (accessible from Lake Minnewanka, close to Banff townsite), Johnston Canyon (on the Bow Valley Parkway, about a third of the way from Banff townsite to Lake Louise), and Mistaya Canyon (off the Icefields Parkway, the road that leads from Lake Louise to Jasper).  Rainy days also can make waterfalls more impressive, which gives a second reason to see Johnston Canyon, as well as visiting Bow Falls in the Banff townsite, or Takkakaw Falls in Yoho National Park, west of Lake Louise.

Another outdoor activity that is suitable for a rainy day is whitewater rafting.  You'll get wet anyway, so it does not matter if it's raining. :-) 

Banff Public Library: Is open to everyone 7 days a week with a good collection of the latest magazines, newspapers, books and internet access.

Still another idea to consider — although it would take at least nine hours — is a visit to the Royal Tyrrell Museum of Paleontology in Drumheller.  It would take three hours to drive there from Banff, at least two hours to see the museum, an hour to have lunch, and three hours to return to Banff.  Drumheller is located between ½ hours and 2 hours northeast of Calgary.  You can avoid Calgary's traffic by driving Banff - Cochrane - Airdrie - Beiseker - Drumheller.  The dinosaur skeletons at the Tyrrell Museum are awesome and well worth a day trip— even in fine weather for that matter.