I've got it!
Drugstores can start selling a little kit, to women. Used within 24 hours after intercourse, it will make a DNA record from any sperm present.
It would be simple and unobtrusive for any woman to take a genetic sample of any man who might impregnate her. If she becomes pregnant, she has what's necessary to prove paternity.
It could be sold right alongside the contraceptives. And of course, if a condom was used properly, there would BE no sperm present, hence no record.
Brilliant!