Welcome. Because of problems with "free dogs" being sold or used for testing, almost no rescue or breeder has free dogs. Even mixed breed owners typically charge a few hundred to ensure the new owner seriously wants the pet. The cost of buying a dog is generally less than it costs to take care of it for a year... and that is if no emergencies arise. The cost for keeping a dog, including yearly vet checks, rabies, heart wormer, flea/tick meds and a decent quality food can easily be $1000 a year. If you cannot afford the rescue fees (typically $200 to 300) or breeder fees, then you need to wait longer to have a good money reserve for buying and caring for your dog. Yes, you can go to some shelters and get dogs free during certain times of the year, but to not spend a few hundred to get the dog you want makes little long term sense.
There are dog shows and basenji owners in your area. Use Tanza's links to find them and arrange to actually meet some basenjis. They are not your typical dog.
Where I am it costs about $2K+ for a dog from a breeder for just about any breed. Goldens may be less because the litters are so large. That may seem like a lot, but, as Debra mentions, dogs cost money to care for. After vet bills and boarding fees and food, it matters little what the dog cost to acquire.
Leaving true puppy mills out of the discussion, given the costs of being a responsible breeder -- testing, vet bills, feeding -- and how much work breeding is, I can't imagine anyone doing it for the money.
That leads to two problems for people looking for dogs. One is that since breeding isn't really lucrative, there aren't a ton of breeders, which means the supply of pups is low. Second is that most Basenji breeders want to place their pups in the best possible homes -- remember they aren't doing it for the money -- which means they're looking for people who have had Basenjis before. Something of a Catch-22.