I'm not sure about tokyo but in the Hokuriku region, namely Ishikawa prefecture- On the Sea of Japan coast- they have things called kazuko-buro.
These are onsens that offer private baths for a short period of time. These places only offer bathing facilites and not accomodation.

If you are after nicer places, there are ryokans that offer private outdoor onsens (ryoten -buro) attached to each room. These are quite expensive, probably about USD400 a night but it would most likely include a large dinner of local specialities. These places are more of a romantic than family-type place though. Also, while expensive, the accomodations would probably not be much to write about, the baths are the main attraction.
I'll be able to post more specific stuff like locations,names and more accurate prices tomorrow if you like.