Yunokawa Onsen (湯の川温泉), over 300 years old, is a nest of relatively built up hotels and spa complexes and has little of the character one might hope for in a venerable old onsen. Yet it's easily accessible from Hakodate by the tram, which makes higaeri (day trips) a breeze.
If you're in Hakodate, be sure to check out the iron-laden waters of the public bath, Yachigashira, which is quite large and has several extremely hot baths and a nice area for relaxing afterwards. Celebs have been known to stop by on occasion, so keep an eye out for tv cameras.