We could start by requiring all domestic cats to be sterilized, so that they don't breed. Then the cat population would be limited to certified breeders and wild cats.
We could also have the same restrictions on cats as dogs. They should not be allowed to be outside unless they are in leash or in a controlled environment (the latter is hard, as cats can climb over standard fences). I guess fully enclosed "cat parks" would be an option.
You may have a harder time training a cat to wear a leash compared to a dog, but it's possible.
It's strange to me that we think cats need to be outside to satisfy their predator instincts, as if dogs and other animals like to be inside all day while we are at work. If someone doesn't like that thought, they have the option of not owning a cat.