Just because you don't see dogs very often in movie westerns doesn't mean they weren't there. It just means that it's difficult to gets dogs to cooperate on a movie set. You need a well-trained dog, and a handler, or they will be, well, just dogs.
Dogs worked on farms, which is where most people lived in the old west. Dogs wandered around towns, sometimes causing problems. Just like today, dogs know who should be there and who shouldn't. If you have ever seen the bad guys sneaking up on the lonely ranch in a movie, consider the fact that they couldn't have gotten anywhere close without the dogs barking their heads off.
So, the next time you see a western movie, use your imagination - and add dogs!