Years ago I had a very laid back bulldog (Snorky). Not an aggressive bone in his body (in his older years he even put up with the antics and ear chewing of a new B puppy)
Long story short, a young man (14) who had been baby sitting his sister's toddler tried and tried to get close to Snork, but he would have nothing to do with him, wouldn't even stay in the same room sometimes. One day this boy was playing with my daughter (then about 7 years old) by swinging her around in circles by her arms, and that dog took after that kid like he was going to KILL him!!
Not for one minute did I think this behavior came out of the blue without provocation. I never again let my daughter play with the babysitter. About 6 months later I saw the toddler and he had bruises all over his little body:mad:, and the brother/baby sitter was in juvie detention.
To this day I'm sure he hurt my Snorky in some way and Snork became aggressive to protect "his pack". He never did it before, he never did it after.
I'm not saying anyone you live with has been aggressive with your dog, but dogs can be very unforgiving when it comes to protecting "their pack". Human behavior, physical as well as verbal can trigger what I feel is the innate protection instinct in some dogs. After that, the perceived threat of aggression, loud noises including laughter and greetings, and swift movements including hugs or backslapping can trigger that innate (or learned) behavior.
Allie, it's just my PPOV but I think something happened that you are unaware of that has set your dog off. Figuring that out may help you finding a way to over come the aggression your Basenji has displayed.
Hope I haven't offended you:o. Good luck, and keep us posted.