They not only survive, they thrive. At least the ones who’s ecosystems are unsullied enough to provide adequate prey. Sure, weaker hunters die off because they either get injured during the hunt or otherwise fail to catch their prey, but strong hunters survive and pass on the genes that enable them to become strong hunters. This is the essence of survival of the fittest. The fittest take calculated risks constantly, and far more often than not come out on top.
They’re picky about what they prey on, typically.
They’ll pick off a sick, old, injured, or juvenile prey animal when they can, which is less risky than a healthy adult in their prime.
While there are cases of predator being severely injured in the process of taking down prey, this is the exception, not the rule, because of how careful they typically are.
For the same reason humans continue to drive cars. It is dangerous, but not so dangerous that the costs outweigh the benefits.
If the number of animals being hurt while attempting to prey is high enough, the species will either die out or it will be forced onto another diet. Palm civets, for example, are relatives of cats that in the distant past used to be predators, but have since adapted to be almost entirely herbivorous. So at some point in the past, one of their ancestors was living in an environment where the costs and risks of preying on other animals were too high, and those felines just started to eat plants instead.