Domesticate Them
This is really your only option. A bit of history lesson on horse domestication: wild horses (here defined as horses in the subgenus Equus (Equus), rather than counting donkeys or zebras) weren't just wiped out in the New World, they were almost wiped out across their entire range in Eurasia as well. Horses are big (producing a lot of meat per individual) and apparently taste good, and so are prime targets for hunter-gatherer societies just like any other megafauna. It just turns out that they are more useful to people as mounts and draft animals than as food.
Humans basically hunted horses to extinction across all of Eurasia except for a few remnant populations in the northern steppes of what is now Kazakhstan and the Caucasus. These northern steppes (including nearby areas like present-day Mongolia) probably held horses longer because it would have been harder for humans to corral or hunt them on foot. A handful of cultures started to keep horses in captivity, mostly for meat and milk (possibly for kumis), and only later found out horses are useful for carrying people or things. At least some of these cultures may be some of the proto-Indo-Europeans (especially given the center of domestication of the horse is the same place that they are theorized to have come from) and their domestication of the horse may have contributed to them being able to spread their cultural ideas everywhere.
These peoples may have domesticated some of the last herds of wild horses, with the species going extinct in the wild shortly thereafter. It was once thought that Przewalski's Horse and the Tarpan represented remnant populations of wild horses, but it turns out that these horses are likely descended from secondarily wild individuals that escaped from captivity thousands of years ago This may not be the only domestication event of the horse, some people have suggested multiple domestication events or even one domestication of the horse combined with new genes added from bringing in wild mares from local populations, but it's very telling that herds of wild horses are pretty much gone after the Neolithic. And that a lot of other wild equid species other than the plains zebra and kiang aren't doing so well.
Notably something similar happened with a lot of Native American groups when horses were reintroduced to North America. A lot of groups (mostly peoples with more sedentary habits) saw horses as most useful for food, but a few of them like the Comanche and the Sioux found that riding horses was a lot more effective (likely because most of these cultures lived on the open plains) and leveraged it into becoming big political powers of the region (Empire of the Summer Moon talks a bit about this).
The best way to keep horses alive in the New World would be to have some culture domesticate them somewhere in either North or South America. Maybe twice if you want horses in both places, as llamas despite being domesticated in South America never seemed to become popular in Mesoamerican cultures like the Aztecs, Maya, or Toltecs, and the rainforests of South and Central America might make horses less useful and therefore less likely to be traded between continents. Wild horses might still exist, but as secondarily feral herds descended from domesticated escapees, similar to the modern Przewalski's horse or the mustangs in North America today.
EDIT: Looking up tarpans a bit more it's possible wild horses survived a bit longer based on historical documentation, though it's not clear whether they were actually wild or descended from escaped domestic animals because historical observers usually didn't make the distinction. Observations seem to be pretty rare. Nevertheless "wild" horses were persecuted a lot because they were perceived as good food and competition with domestic horses for grass, and they only did well where large groups of people generally weren't.