We regularly travel to The Black Forrest region and have stayed in Freiburg, Bad Kronzingen and Bad Peterstaal. If you want a city break, go for Freiburg.
If you are staying within the region for more than a couple of nights then the hotel will issue you with a KOnus Card which will give you discounts off various attractions as well as free local public transport. Basel is expensive and, depending on where you are staying, either use Basel or Stuttgart airports or even take the train from these airports. Basel is easy to get to by ICE from Freiburg and you could rent a car from the city centre and avoid paying the airport charges.