The "correct" way to deal with it is the use of natural predators.
Problem is people often don't like or know how to live with proper apex predators so they get killed off and we have to artificially control animal numbers.
However, in many places, we have introduced non-native species that never had natural predators. Then what?
I can't help but thinking of the Simpsons episode "Bart the Mother"
Principal Skinner: Well, I was wrong. The lizards are a godsend.
Lisa: But isn't that a bit short-sighted? What happens when we're overrun by lizards?
Principal Skinner: No problem. We simply unleash wave after wave of Chinese needle snakes. They'll wipe out the lizards.
Lisa: But aren't the snakes even worse?
Principal Skinner: Yes, but we're prepared for that. We've lined up a fabulous type of gorilla that thrives on snake meat.
Lisa: Then we're stuck with gorillas!
Principal Skinner: No, that's the beautiful part. When wintertime rolls around, the gorillas simply freeze to death.
Lisa: Hmm.
EDIT: Yes, I understand you are referring to keeping predators around, and not necessarily introducing new predators. But this is still what jumped in to my mind.
I think adding another animal to the situation would only exacerbate the problem. Let's say we add lions. Then not only goats are eaten, but other animals.
Hunting goats to local extinction is the cleanest and most humane way to deal with them.
Problem is people often don't like or know how to live with proper apex predators so they get killed off and we have to artificially control animal numbers.
However, in many places, we have introduced non-native species that never had natural predators. Then what?