I know that the first traffic light was in Britain in 1866. I am not aware of any laws regarding riding a horse or driving a carriage while drunk, at least in North America, prior to the 1910's.

I also read rumor that President Grant was arrest for speeding while drunk and that President Pierce actually struck someone. I imagine getting struck or trampled by a horse or team of them would be mighty unpleasant.

If you have ever been in a carriage you see that horses do not react to steering like a car does but they mostly follow the path or road on their own, especially if they know the road. So getting off the road or crashing into something would not be such a problem when driving drunk, as horses normally try to prevent damage on themselves. But another thing was obviously, depending on the personality of the driver, that he would do something that would put the horses to unrest like giving weird commands or making loud unexpected sounds. This would make the horses go wild. I don't know if there were laws elsewhere in the world, but for example the first common laws for roads were introduced in Germany in 1909, so before the rise of the automobile no one saw the need for laws regulating traffic even for cars.