I just did some searching, and I couldn't find a single reference to any law criminalizing drunk riding that isn't fairly recent.

Looking at some of the state laws criminalizing drunk riding: Minnesota only criminalized it in 2000, Oregon explicitly extended the motor vehicle code to riding animals, and it appears several other states(NH,SD,MA) it was the courts determined that horses were considered vehicles under the law. Interestingly SD explicitly excluded horses from the definition of vehicles back in 2006. I didn't look at every state, but from those laws and a few articles I found about the earliest drunk driving laws(this one about Vermont is interesting) it seems likely that the constable was wrong.

I think the reason drunk riding probably wasn't illegal before cars is that it didn't really need to be. Horses aren't machines, they don't do exactly what the rider wants them to do at all times. If a rider is completely wasted and passes another horse or pedestrian, the horse will instinctively avoid them even if the rider directs them toward collision. It really only becomes an issue when 2000+lb machines zooming around that have the potential to scare the horse that the rider needs to be fully capable of taking control.

I don't know why I just did an effort post about this, but I found it interesting.