The Ranger Station building was built in 1892 (at right, Managing the Matchless Wonders, link)
This 1912 Map shows the location of "Riverside Station"
The Riverside Ranger Station was located 4 miles east of the boundary of Yellowstone National Park, on the right side of the road, near the ford of the Madison River where the old Gneiss Creek trail (Riverside-Willow Park Trail) and Madison Plateau Trail met. (right, Haynes Guide, link)
This map shows where Riverside was roughly located. (Yellowstone Rules and Regulations 1920, link)
This 1923 USGS stream gauge description (at right, link) describes the gauge's location, in front of the Riverside Ranger Station.
The location of the Riverside Ranger Station appears to move further West at some point, ending up in the area of The Barns (aka Christmas Tree Park). This map (at left) is from 1958.