Clubs that spring to mind are:
- Centrepoint: http://www.centrepoint.ch

Also, the Centrepoint "links" page has links to some other clubs like the American Women's Club, Anglo-Swiss club.

If you like amateur dramatics, have a look at the Semi-Circle ( http://www.semi-circle.ch ) - they have regular stuff going on. There is a production currently in rehearsal but it isn't too late to get involved backstage. Also, they have regular play readings - also a good way to meet folk.

If you like running, drinking beer and having a fun time (with emphasis on the latter ) check out the Hash House Harriers - http://basel.harrier.eu.org.

By far the best way to meet Swiss folk is to get involved in a club for your favourite sport or other hobby. I'm into Aikido and joined a local Aikido club near to where I live and they've been great with me. Also now I am involved with a Fasnacht clique which is about as "going native" as you can possibly get.

If you want to ask some Basel-specific questions, I'd suggest joining the following two Yahoo groups:

http://groups.yahoo.com/group/BaselLounge
http://groups.yahoo.com/group/BaselExpat

Do take advantage of the "Search" function in the Yahoo groups as there are quite a lot of useful postings on there.

HTH.
Nick