I remember when the remuda was operating in the Basin. The stables were located near what's now the entrance to the employee residence area in the Basin, just past the turn-off to the group campground.
They offered two rides, a half-day to the Window and the full-day to the South Rim. They would go up and down the Laguna Meadow trail. I don't remember the route they took from Laguna Meadow to the South Rim, but they did use the trail from Boot Springs to the Rim (or vice-versa).
Needless to say, the water quality and trail conditions are much better now that there's only an occasional horse.