Basel is a great place to come with kids, in my opinion! The BCT is a fabulous resource for mums and kids, www.baselchildrenstrust.ch
its an organisation mostly organised and run by expats and their young families. It organises lots of playgroups, meetings, nights out and fun events.
In terms of places to stay, you don't have to travel far, all the surrounding villages have a selection of shops and quick access to the countryside. We are in Oberwil, its 10 mins by public transport to the city centre, 20 mins for my husband to cycle to work. But we also walk out the door into forests and fields which are full of dog walkers, cyclists, joggers etc. Its a really safe and peaceful place for families. Other nice spots are Riehen to the north of the city, Muttenz, and then getting a little further out of town Reinach, Aesch, Arlesheim, Biel Benken and Therwil are all well connected to the town and have lots to offer. There are a lot of expats with young kids here so lots of opportunity to meet others and hook up with mums of kids the same age.
If you have a car then Germany/France for cheaper shopping is no more than 20 mins away. Riehen is much closer to the border. If you anticipate being at the INternational Schools (I know it is a way ahead) then you might want to live closer to those. One car families are quite normal, workers use the excellent public transport system to get to work and usually the mums end up with the car to do the shopping and child related activities!
Hope that helps a bit, feel free to ask more if it doesn't make sense!