Last year I went to Basel on my own for a month, which in hindsight was probably a bit too long. In the first week I visited different parts of the city by tram every day, sampling the culture. The centre of the city itself really is beautiful with some stunning architecture. The National History Museum is well worth a visit, as is the Münsterplatz square near the cathedral. After a week or so it turned out I'd done pretty much everything there was to do in Basel and the weather took a turn for the worse. I then spent most days in my hotel room simply watching Swiss TV (which by the way is dire). At nights I stopped eating out and if I wasn't crying myself to sleep I'd be wanking furiously into a sock.
So if you decide to go to Basel I would recommend you do it for no longer than a week, maybe ten days max.