One thing about Switzerland, you can stay in one place and still see a lot of it.

Schaffhausen - the Rhine Falls and St Gallen - beautiful monastery and its library founded by Irish monks would be not that far to the east.

Zurich & Berne are quite close too, though are rather much like every other big city other than the historical centres.

Haven't stayed in Basle so can't make any suggestions about cheap (if it's available)accommodation.