Welcome to Switzerland Andie!
I live in the Zurich area so don't know much about Basel, but you could go here and ask...
http://groups.yahoo.com/group/BaselParenting
There are more than 200 members there so someone is bound to have some info for you:-)
Here are some links to things to do there with your kids in that area...
BASEL Dampfbahn - Mini Steam Engine - Psychiatrichen Universitaetsklinik Basel Mini Steam Engine rides every second Sunday (middle of April through middle of October). http://www.dampfbahn-basel.ch Tierpark Lange Erlen - Basel Animal Park http://www.erlen-verein.ch Zoo - Basel http://www.zoobasel.ch Wiesner's Beeriland - Bottmingen Berry picking in June & July. http://www.beeriland.ch Park "Im Grünen" - Münchenstein
Playground, Mini-Golf, Restaurant...
http://www.seegarten-gruen80.ch
Solarbob - Langenbruck (BL)
A rodelbahn (alpine summer sled run) that uses solar energy for the lift.
http://www.solarbob.ch