I agree with others about the BCT. I am a coordinator at one of their playgroups,(though I am due to leave Basel next week

). the BCT is a great starting place with playgroups for different ages from new born to school age. You will connect with other English speaking parents who have been here for varying times, short and long and will most likely have been through whatever situation you may find yourself in, and have advice! Check out their website
http://www.baselchildrenstrust.ch/
Enjoy Basel. It is a great place to be with kids, and if you are just arriving, it is a wonderful time to arrive - lots goes on over the summer in the city.