There is a woman on B-pix, has 2 basenjis and 2 parrots, the birds run around on the floor with the dogs, the dogs respect them.
Many years ago we had 3 basenjis and adopted a neglected peach-face lovebird from a neighbor. It was just a "cage bird" and we got it a big tall, round cage that the basenjis could sniff and see. At first they wanted the bird really bad, but they got used to it and after a few months we opened the cage door and let the bird run around outside the cage. The dogs tolerated the bird, then would let the bird bathe in their water bowl. The 3 basenjis and 1 lovebird would all be around a big bowl eating oatmeal, it was very cool. We were so relaxed about "Green puppy" that we didn't think to lock him in his cage when another basenji came over to play…and that was the sudden end of Greenpuppy. I couldn't be too mad, it was a quick natural reaction for a dog, but we cried and buried him in the yard. I still have one of his feathers in a baggie in my freezer, and that was about 20 years ago, sniff, sniff.
So to make a short answer long, some basenjis can learn to love a bird or two, but you have to be very patient. And other birds have too strong a prey drive for them ever to co-exist free together.
Anne in Tampa, with 3 basenjis and no birds