We day-hiked in Banff for 4 days in the middle of August with our 5 month old. (Iíve written about this some more at https://alocalminimum.wordpress.com/
A few comments and then brief descriptions of the trails we did, which I think are good examples of trails to do with a kid this age:
1) At 5 months our son is still in a front carrier. I think it's possible to do more or less the types of hikes you would do on your own if you're still in the front-carrier stage, but they have to be *much shorter* because a) it's always slow getting out in the morning with a baby, b) you need to take many more breaks for the baby to feed, be changed and stretch and c) you probably don't want your baby sitting immobile most of the time for the whole day.
2) Assuming you are going as a couple with one parent carrying the baby and one the knapsack with food, water, and layers for the three of you plus diapers, first aid, camera, cell phone etc etc. it's important to pack smartly: we found that the weather reports were pretty accurate and by checking them right before heading out we were able to choose only the layers we needed. We also tried to be realistic about how many diapers we actually needed and not go overboard.
1) Johnston Canyon to Ink Pots: ~11.5km out and back (~5.5 hours).
This is an easy and extremely crowded hike. The canyon and waterfalls are nice, but very crowded. Definitely continue to the Ink Pots - very beautiful and the trail is less busy.
2) Sunshine Meadows: ~10km loop(s) (~6 hours+gondola).
This is the hike we did that I most recommend, it is easy-moderate with some short ups and downs. It is relatively (for Banff) not crowded although the gondola is pricey.
The 25 minute gondola gets you up into the alpine with no effort and it (and the chairlift!) is a fun ride. The Standish viewpoint from the top of the chairlift is breathtaking as are Rock Isle, Laryx and Grizzly lakes and the meadows themselves. The view from Simpson viewpoint at the back of the Laryx loop has been burned out. After completing the loop we hiked from Twin Cairns junction to the Monarch Lookout and then back to the gondola on the meadow park trail. This adds a nice section of alpine hiking to the trip.
3) Moraine Lake - Eiffel Lake: ~13km out and back (~9 hours)
The Lake Louise and Moraine Lake area is insanely busy - take the 7am direct shuttle from the Park and Ride to Moraine Lake.
Take your time exploring Moraine Lake - the rock pile, and the walk along the shore - and taking pictures. It is extremely busy but for good reason.
From Moraine lake, the trail to Eiffel lake starts with 2.5km of easy-moderate switchbacks before leveling out for 3.5km of easy, flat walking. This trail is also very busy, but the further you go the fewer people there are. Some of the walkis forested, but most of the trail offers stunning views. Eiffel lake at the end is beautiful - make sure to walk down from the trail to the lake itself.
At the end of the day, when things were slowing down, we drove up to Lake Louise and spent some time taking in the views.
4) Vista & Arnica Lakes: ~11km out and back (~7.5 hours)
There is an easy ~1.5km hike down to Vista Lake and then a difficult ~4km uphill hike to Arnica Lake. We almost gave up, but as we were considering whether to turn around after 5 hours, we saw the first other hikers on the trail who told us we were only 20 minutes below the lake.
This was the most difficult, but by far the least busy, of the hikes we did. We only saw 3 other pairs of hikers until we started heading down, and they were all near the top. On the way back, we passed maybe 5 or 6 groups that were going up. The lakes are beautiful, there are some open views and a lot of wooded areas, and the solitude definitely makes this hike worth it.