I'm a huge fan of BRAT and their associated rescue groups, but it is possible to find a rescue basenji through other sources as well. We have been in touch with our local (well, 4 hours away, but our closest) official basenji rescue for a long time, and try to donate to them now and then, but nothing ever quite worked out for us getting a pup from them.
We found our girl Fiji through a Pennysaver ad, placed by a woman looking for a new home for her. Our second guy we found on Petfinder - he was in a shelter a couple of states away and it took a lot of logistic maneuvering to get him home, but he was determined to be with us :p
This process is not as clear cut as using BRAT. Petfinder lists an amazing range of dogs as basenjis - pretty much anything reddish in color or with pointy ears. :rolleyes: And some that aren't even that similar. But if you're willing to check their website regularly and slog through a lot of listings, pure basenjis do turn up in shelters.
If you find a pup being privately re-homed, you can ask a lot of questions, and they may even give you truthful answers regarding health, disposition etc. Shelters have a hard time providing much info, because they generally just don't know. A visit can reveal some things about temperament. Of course, basenjis seem to be clever enough to be on their best behavior until they have time to make you fall in love…then, look out!
Unlike BRAT, where dogs are fully evaluated and you generally know a lot about what you're getting, shelter dogs may have a lot of challenges that surprise you. And that's even trickier when it comes to making sure dogs are compatible with kids, so be careful.
Our stories have had happy endings. Well, they're on their way to happy endings, anyway. Our new guy Bongo has been a little more challenging than our girl, but he's coming along. Of course, we have no idea what lies ahead health-wise for our kids since shelter dogs are not likely tested for Fanconi or other problems, or if they have been that info is not available.
The other thing you might consider is breeders near you that have adult dogs they are retiring from showing/breeding. http://khanisbasenjis.com/AvailableBasenjis.html
That would be a safer bet, since again you'd have a lot of info on the dog's temperament and health.
Good luck getting a basenji back into your family. Hard to live without them once you've gotten to know them, isn't it?