Savoie/Aosto while nice is quite off Basel actually
From Basel the Black Forrest is just around the corner, for a winery the Alsace is not far as is the Kaiserstuhl area on the german side, quite nice for hiking but probably quite warm/hot in August, an hour from Basel north for both.
Going south from Basel some of the majestic alps are just like an hour away … places like Lake Lucerne / Engelberg would be worth a visit.
In Basel a stroll through the pedestrian area city center is quite nice, some shopping for the wife included, a good set of museums in Basel actually too.
Going north you hit the Black Forrest, highest place would be the Feldberg, nice for hiking especially in August. Lots of nice places to visit, castles, great nature like the Wutach-Schlucht, geourgous place to hike through and can do from short trips to longer ones.
Down the Rhine valley you hit Kaiserstuhl and Freiburg, lovely city, worth a visit too. Good area for hiking, wineries, good food, shopping. On the other side of the Rhine in France you have nice places like Colmar, or little wine villages like Riquewihr.
Up the Rhine to the east you would be hitting the Rheinfall and further the Bodensee if you like water more than the mountains.
Overall, great areas for short sight-seeing trips, all of them actually, recommended, and well, its where I live so hit me up with further questions when getting closer.