Will you be visiting other places? I ask because very close to Basel is one of my favorite cities in Europe, Luzern. If you are not already going there, you could consider a day trip, but I would want to spend at least two days there. One to visit the old town/river areas and another to take an excursion to the top of a mountain (Rigi, Pilatus or Titus).
I really liked Basel. "Don't miss" for me would be the (1)Muenster and be sure to catch the Rhine view from behind it.(2)Rathuis and the market square in front of it. I would go morning to midday to be sure to catch the market.(3)Meander around the very colorful old town area and (4) Cross the river by the bridge by the art museum and walk to the bridge that crosses back into the old town.
Basel has a wonderful website, Basel.com. It has wonderful ideas. I contacted them and asked for maps of walking tours and they were happy to mail them.