Antinatalism doesn’t only regard human life. We should also think about animals (hence the rise of the vegan movement, too).
Anyway, I think if you must own a pet as an antinatalist, then the ethical thing to do would be to rescue a pet that was given up by its previous owners. When you rescue a pet, you save them from dying lonely in the shelter. It’s a great thing to do. Don’t pay breeders to breed more cats or dogs as that is exactly what antinatalism stands against: Creating life only for them to potentially suffer a terrible fate.
Sisyphus Shrugged 