Strasbourg is beautiful and 1.5 hours from basel by car, but not necessarily on the cheap side. Cheaper than Zürich definately though.

Konstanz is reachable in no more than an hour and is a really nice place too. You've got the huge lake of Konstanz, (possible evening dinner boat trip or good long dog walking!) very reasonable good quality restaurants (Large steaks for equivalent of 20 francs, imagine that!!) and super cheap shopping with the opportunity of claiming tax back on certain things when you cross the border back into Switzerland (check other threads for the ins and outs, do's and dont's of tax back etc) There is also at least one wellness spa within a 15 min drive of the city.