Well, they may have representatives on some shoots, such as on the New Mexico set of James Mangold's 3:10 to Yuma remake, but while they can prevent intentional cruelty to animals, they can't keep the occasional accident from happening. On Monday, there was such an unfortunate event during the filming of 3:10 to Yuma, as a horse and its rider collided with a "camera-carrying vehicle", causing enough injury to the animal that it had to be put to death (the rider went to the hospital in critical condition). Beginning Wednesday, a whole team of AHA personnel were on site to conduct an in-depth investigation and request a necropsy in order to find out if there were any non-random factors involved.
Certainly with few westerns being made these days, accidents involving horses are kept to a minimum, but last year two horses died during the filming of the new film Flicka. I guess the only thing to avoid future tragedies would be to feature only CGI animals.
To keep up with the AHA's monitoring and review of animal safety in new releases, visit their Film & TV Unit page.