Asama Onsen is the closest hot spring district to central Matsumoto — about 15 minutes by bus from the station. The springs were noted in the eighth-century Manyōshū and have been a bathing resort since at least the Edo period. The water is mildly alkaline and the bath houses range from very basic public baths to upscale ryokan with rooms.
For a day visit, the public bath Hot Plaza Asama is the easiest option, with separate male and female baths and a small outdoor pool. Stay overnight at one of the inns to do the place properly.