Man, this turned into a novel.
I owned one, he was pretty awesome. It was already mentioned that they don't bark, they yodel, but honestly they don't yodel unless they're distressed. I never could get mine to yodel on command. When they do start yodeling though, holy shit it sounds like they're being molested or something. Especially if you try to crate train them as puppies.
They're small dogs and will easily live 20 something years. I've heard they are the original terrier, that pretty much any other breed that's a terrier came from this breed. I'd never heard they were bred to hunt lions, I heard they were basically one step away from wild dogs.
They often get listed low on the breed intelligence charts, but that's not because they're dumb, it's because they're hard headed. You don't really train a basenji, you have to convince them that they want to do what you want them to do. Traditional carrot and stick approaches don't work (at least the stick part doesn't). For instance, I used to let my dogs out to take a lap around the property when I got home. In the winter sometimes it would be dark when I got home, so I wouldn't let them out. The basenji started shitting in the house at night. I punished him for it, but figured it was because he liked to shit out in the neighborhood, didn't wanna shit in the yard, and couldn't hold it all night. He was actually pretty good about not shitting all over the yard, that's another rep basenji's have. He kept it up, and I starting making him stay out all night. So he'd wait till I let him in in the morning to feed him, and go shit in the house while I was showering. He only did this on days when I got home too late to let them out. The fucker was shitting in the house to punish me for not letting him out. Tell me that's not a smart dog. And he would stand by the shit all defiant too, not slink away like the other dog when I got upset.
They are escape artists. I read from one owner that a chain link fence is nothing more than a ladder to freedom for a basenji. After my shepherd died, mine turned into a real escape artist. As long as she was alive, he only got out when there was a hole big enough for both of them to fit. My yard was fenced with range fencing. If you don't know what that is, think of chicken wire, but it's a lot thicker wire and they form rectangles that get larger as you go up. At some point he figured out he could climb through the fence about 4 levels up. But I guess it hurt because he actually went on a fucking diet so he'd be thin enough to crawl through the square 3 levels up. A self imposed diet, all of a sudden, he was only eating half his food. Got worried and took him to the vet and everything. He was in and out of my yard every day for I don't know how long. He'd always be back in the yard when I got home though. I only found out because he started ranging out to the highway and the dog catchers finally figured out where he lived and left a note on my door. They said they saw him jump through the fence at a dead run. Once they tipped me off to it, I put him in the yard, went outside the fence and watched him do it.
They are aggressive towards other non dog animals. You can have things like cats and ferrets, but you have to watch them closely around each other until you're sure they've been accepted into the pack. Mine hated kids too. He split my nephew's lip open once, but in fairness, my nephew was being fucking stupid and it was a lesson he needed to learn. If a dog is hiding from you and growling and baring it's teeth and the dogs owner is telling you to leave it alone ... leave it the fuck alone no matter how cute you think it is. That seems to be common for small dogs though in my experience. Big dogs generally don't seem to give a fuck and can chill out around kids. Little dogs just can't. Probably because to them, that uncoordinated thing grasping at them is still a fucking monster.