I took a look at the Basenji Puppies (for sale). I was happy to see that health testing and other things are requested, but really most the dogs are not even in the USA. Like Pat said, please go to the Basenji Club first. There are also breeders with older dogs (ie not puppies) that might need homes and rescue. The one.. and it is a biggie.. concern is that any dog you get– be it breeder or rescue, if they do not have Fanconi testing, don't get the animal unless they test. If you find a breeder or dog, please let us know so we can help you verify actual testing if you need help.
Now the really bad news. Basenji are not hypoallergenic. NO DOG IS. It is a bunch of hooey spread by people without real information or breeders pushing their dogs. While lower shedding dogs do help with dander, that isn't the same thing. You might actually see about fostering a basenji for rescue and see how your family reacts allergy wise.
Now for the really really really bad news. Allergy concerns are minuscule part of breed selection, especially with a basenji. Sorry but they are not the dog for most people and they most certainly are not the dog for someone not familiar with them that is just looking for a less allergy inducing breed. There are continuous growing hair type dogs that shed less and have far nicer temperaments. Please manage to actually spend time with and around the dogs before you move forward.