I agree with eeeefarm - if you don't want your basenji hunting, then don't let her run around unsupervised because she is obviously a great hunter. Given enough time, I am guessing the local wildlife will learn to avoid your property (or she will kill them all off).
If the critters are coming into your yard and it is not huge, perhaps you can use a product like NIMBY to deter the critters from coming in.
My Basenji has caught multiple rabbits and squirrels while on a leash as well. I went to using a shorter leash. At the cottage I have her trained to stay on the property but I do put a bell on her collar. The bell is partly to let me know when she is chasing something so I can ensure she doesn't leave but it's also to let the squirrels know where she is. They tease her relentlessly so if she gets one; it's dead!
If he/she ate any of the rabbit, get some medicine for tapeworms
I learned that you seeing their tapes is how the vet knows they're there. After a couple of years, all I had to do was call the vet, say the dogs got another rabbit, and the medicine would be ready for me to pick it up.
It's what Basenjis have done for some 6000 years !
These are hunting hounds and hunting is hot wired into their DNA.
All mine have hunted, killed rabbits, squirrels, pheasant, and, in a neighbouring chicken feed store, rats. If I find them here in the garden for example, I do try to take the carcass away.
Interestingly, a rabbit's head is always the first thing on the menu. I used to find headless carcasses in the shrubbery from time to time when Ziggy and her daughter, Hope, were alive. Ziggy caught birds too, as they flew out of the hedgerows. She was so quick.
One of mine, living with an older Basenji, would catch rabbits and pheasants and drag them across the fields and lay them beside the old girl when she could no longer hunt. Owners found this very touching but could have done without all the fleas she brought into the house.
So far Mku's tally is two squirrels. But he didn't eat either of them. Not much meat on a squirrel.
If you don't want them to hunt, I suppose you have to keep them tethered. But this behaviour is to be expected in a hunting hound.
I was going to suggest bells on the collar, as many have mentioned. I have an enormous bell on doodle's safety orange harness. I consider it fair warning to any critters! The bell allows her free roaming in the woods and I can keep tabs on her through the jingle jangle. She hasn't caught anything (yet), but she doesn't typically get the opportunity.
Jengo and Logan both chased or chase prey. Both are a bit awkward, loud and reactive. Sparkle, on the other hand, is much more stealthy, focused and patient. I did see Jengo nail a mouse once. Neither Logan nor Sparkle have nailed one while here with us... yet.
My biggest concern would be them getting injured during an attack. A squirrel biting them. A raccoon could be really bad. We also have wild coyotes not far. All that said... the head wants what the head wants. No idea how to stop that.