My son and I booked a one-hour horseback ride at Sacred Monument Tours, whose stable is located just beyond the cabins at the View hotel, from 10-11 in the morning. Robert was our excellent Navajo guide, and our mounts, Willie and Ecker, were experienced trail horses that were perfect for a couple of inexperienced riders like ourselves (though other mounts and longer rides are available for more experienced riders). The ride took us around the base of one of the Mittens and back, and was the fulfillment of a lifetime bucket list for someone like me (68yo) who grew up on John Ford westerns.
As an added bonus, as Robert pointed out, the Sacred Monument stables are located at the site of the Edwards family homestead in The Searchers. Standing there, one can see the exact trail that Ethan (John Wayne) rode up to the homestead in The Searchers' opening scene!
All in all, highly recommended.