Most people had horses, they were extremely common.

This is, simply, not true - except in movies and TV. Horse ownership was expensive and it could be annoying because of the very problem you are mentioning. The West was (and is) a very urban place, and having a horse in an urban setting makes little sense. I discuss this in one of my books about Virginia City, Nevada (2012): the archaeology of that important industrial-mining center yields surprisingly few horse-related artifacts. That's because people tended to walk within the city and to take "mass transit" (i.e. trains or stagecoaches) when traveling elsewhere. There was no need to shoulder the expense of horse ownership when these two options (i.e. walking and mass transit) were available.

Horse ownership was mandatory for people who lived in a rural setting, but given my first point - the West was a very urban place - rural living (the classic rancher and cowboy) was relatively rare. In addition the wealthy sometimes indulged in horse ownership in an urban setting because they wanted to have a carriage for show or convenience.

So ... given that context, to your question: what did people do with their horses when they took trains and stagecoaches (or more to the point, what did wealthy people do with their horses while living in cities)? The answer is that there were urban stables. These were "parking garages" that offered the full range of services: you would have to pay to have your horse housed, fed, and exercised. It was expensive, which is why horse ownership was restricted to the wealthy or to those living in a rural setting who had no choice because they did not have access to mass transit.

If someone rode to town to take a train or stagecoach and was going to be gone for a long time, they would be forced to sell their horse in the city or they could be taken there by a friend, much like occurs when someone takes an airplane for a prolonged trip: airport parking fees can be expensive!

For your convenience, I just happened to have posted (for a university guest lecture) the chapter from my book that addresses the question of horse ownership. Go to page 60 (and the pages following) for more discussion about your excellent question.