There were urban stables. Of course, most travelers took mass transit (i.e. stage coaches or railroads), so they arrived without horses. Most people of the West did not own horses (contrary to the mythos and the depiction in movies), but for those who did and who arrived either on horseback or in carriages of some sort, urban stables were available to provide care for the animal(s).

I deal with this in my book, Virginia City: Secrets of a Western Past (2012).