Just thought I'd put my 2 centimes in as well - although my experience is a bit limited. We currently do the live in France and commute thing - well, my husband does, I'm not yet working, but hopefully that will change! The commute is made really easy by great public transport links, and we love living in France and the lifestyle here, yet still have access to all the services in Basel. We get the best of both worlds! If you are looking for some information on schools close to the border area, there is a link in the St Louis regional page here
, although it is only in French. My husband has a colleague who moved to France with 2 kids, and while they knew no French whatsoever before they started, it didn't take them too long to settle in and get a handle on the language, and now both kids are top of their respective classes.
It is worth looking into the Swiss school system thoroughly, and at what the schools here is Basel offer, as I have heard that students can choose to take some of their courses in English - I don't know at what level this starts, I'm afraid I can't give you too many more details, as I am in the process of checking this out myself, but it is worth investigating.
It is also worth checking out your work contract and what provisions your company give you in there for schooling - sometimes companies will give help with fees, sometimes they will pay all fees, and in that case, the international schools could be an option.
In any case, we have found the borderlands - both Basel and France - a great place to live and work.