The closest natural hot springs to Rocky Mountain National Park are at Hot Sulphur Springs, CO, about 20 miles west of the Kawuneeche Visitor Center on Hwy 40.
Lots of history here - in operation since 1864, you can soak where the Ute Indians once bathed. It's totally natural - over 200,000 gallons of fresh hot mineral spring water flow through the pools daily, no chemicals or filtering.
The 24 pools are open daily, year-round from 8am to 9:45p. No pets are allowed on the property, and kids under 12 can only go in four of the pools - the rest are Adult Only to keep the experience quiet and relaxing. Fees charged.
Minerals in the water are good for hair & skin (sodium, sulfates, chloride, magnesium, potassium, calcium, floride, lithium & zinc), and you'll leave after your soak feeling oh-so-refreshed.
Further west and north of RMNP in Steamboat Springs, CO, but still within day trip range of RMNP there are two other popular hot springs: Strawberry Park Hot Springs in the mountains above town, and Old Town Hot Springs right in town.