Speaking only from my own perspective as a "pet owner" of rescued dogs:
AKC is the real, official registry used by responsible breeders, other registries are "knock-offs", invented so pet stores and puppy-mills and BYBs can sell "registered" dogs, as I understand it. The only reason you NEED to be registered is to compete in conformation, lure coursing, showing or agility to get championship points, or to start a breeding program yourself (after reaching those championship goals in some of the aforementioned activities).
If your basenji is "just" your pet, love of your life, best friend and roommate, having them registered isn't really important. Shelter rescue basenjis and many others in rescue don't have papers but are purebred basenjis. Once neutered it doesn't matter as you can get the ILP certification to compete in all activities except conformation.
The really important thing that hopefully comes with the AKC responsibly bred dog is health testing and healthy parents and grandparents. You probably won't have that with the knock-off registries, but you can do it yourself for your own peace of mind, especially the Fanconi testing. There really isn't much reason to do some of the other testing, as hopefully no one would breed a dog without that deep background knowledge of genetic health and soundness along with great conformation, the things that make a dog worthy of having their genes passed along. There are so many thousands of dogs, including basenjis, euthanized every year that need homes so there is no reason for anyone to breed dogs that aren't healthy and genetically sound!
Thank goodness there are responsible breeders that supply us with healthy pets and show dogs, and if ever the shelters empty out and there are more waiting homes than there are dogs, then the good breeders can pick up the pace! But I'm afraid that for the near future, there will be enough basenjis and other breeds supplied by BYB and shelters to fill all the available homes.
So enough of my soapbox, sorry, once I get going it's hard to stop. Love your dogs, register them with AKC or neuter them and get an ILP and go for some agility or Rally, obedience, or coursing, or just take long walks with your beloved basenjis and enjoy them for the unique 'people' they are!
Anne in Tampa