By the honor system, mostly.
Stallion owners submit a breeding report at the end of the year where they list all the mares bred to this stallion, and the dates. The document states that the breedings were done by live cover, and they sign this document.
The owners of both the mare & stallion sign the registration application, stating that the foal is the result of a live cover breeding.
And that’s really about all – signed statements by the owners that it was a live cover.
But there are plenty of ways to investigate if suspicion arises.
Mares are typically bred several times over a period of a few days, generally at the stallion’s farm. There would be records of the mare arriving & departing, records from the transporters who hauled her there, the stud farm’s vet usually examines a mare for diseases before breeding, and the vet’s office would have records of this, most stud farms keep a daily breeding diary, etc.
Some of these could be faked, but that gets harder as there are more of them, or they involve records at other places. Financial records, like receipts from the gas station where the transporters filled up gas, or checks to the vet, etc. are hard to fake.
And suspicion could arise easily. This isn’t really something that would be done by a single person, there would be several people involved. From stable hands at both the stallion & mare farms to the FedEx guy who delivers the distinctive liquid nitrogen cooled container of semen. Counting on all of them to keep this secret forever is unlikely. Especially if the foal goes on to become a famous racehorse. There is a lot of gossip in the racehorse breeding business, and something like this would be a really juicy item. It’d be very likely to slip out.
And the Registry has very extensive powers to investigate this – much more than a Court of Law. To start with, you have the burden of proof to prove that your foal is eligible to be registered. If you don’t cooperate, the Registry can suspend your membership, meaning all your other foals couldn’t be registered either. And other horses owned by you might not be allowed to race. The Registry takes such allegations very seriously, and investigates thouroughly.