Two places to check would be at Sea Horse Ranch and a place across from Coconut Palms in Cabarete that used to offer horseback riding.
I am not sure CocoNut Palms (on Playa Encuentro, in between Sosua and Cabarete) would qualify for a universal recommendation. I reckon it depends on what kind of horse back riding experience one is looking for.
I've just recently seen these "horses"... some are barely 2-dimensional, so underfed and small. The "guy" who presented the horses, a local "kid" had no clue whatsoever about horses and the prices for an hour were quoted somewhere around USD 30.oo or more (it was so ridiculous for the "quality", I did not really concentrate on remembering exactly).
It might be something "tourists" who are not really into horses might want to "do" as part of their leisure time here, as they also offer romantic moonlight beach trips... but horse lovers may be appalled by the condition of the animals and the saddles etc.
Some owners at SeaHorse Range have some "serious" horses... I don't know which ones are available to the public and if they are exclusively classical riding or also some western (pleasure, trail) trained horses.
I understand there is another center nearby which had been recommended years ago... but the name escapes me right now. I haven't visited them either, so I can't comment.
Sadly, compared to much of Latin America and especially Mexico, Colombia, Peru and even Ecuador, the DR has virtually no sizable Equine "culture". I miss that.
... J-D.