Brennannbooth... I live in Basel, and have recently resigned to become a SAHD. (I asked for opinions here)
We did the whole nanny thing while in the UK and tried to do the same over here. Unfortunately we ended up feeling it better that either DW or I stop working because we were unhappy with the available childcare.
To try and answer your questions:
What's it like?: Fantastic, in many ways. There are cultural norms over here that take some getting used to, but nothing too challenging. Basel is definitely one of the more child friendly cities, the cultural life is very active, the ex-pat community is large and welcoming.
Unusual? No. Being a SAHD over here is not that ususual, and there's a Basel SAHD group. I'll be tracking that down when I finish working. There are ex-pat curry nights organised through www.englishforum.ch, and several other opportunities to make friends both as a couple and as a stay at home parent.
Are you thinking of living in town or a little further out?
Oh, and this is a question you'll probably be asked by everyone if you come to Basel: "Hi! Are you with Roche or Novartis?"