Private rooms used to be the best places to stay in Karlovy Vary with regard to quality and price. But this is changing as more and more hotels renovate and raise standards -- as well as prices. Private accommodations can still provide better value, but they take a little extra work. If you want to arrange a room, try the Infocentrum. Expect to pay about 1,500Kc ($71/£36) for a double.
Some of the town's major spa hotels accommodate only those who are paying for complete treatment, unless for some reason their occupancy rates are particularly low. The hotels I've listed here accept guests for stays of any length.