As previously stated, basenjis are not hypoallergenic. Not only are shedding and dander a concern, but also saliva. Some people have terrible reactions to being licked. Whether a basenji is a good "first" dog depends on many factors and is why a good breeder or rescue group is invaluable in the decision of who gets which pup. Shelters do not screen homes for compatibility or appropriateness.
It sounds like you are a minor, living in the home of your parents (or at least a mother). That means you have no control over whether or not you will be able to keep a dog if/when the head of the household decides he/she can no longer tolerate living with a dog. The decision will be out of your control. I would recommend you wait until you have complete control over your living arrangements and have the right circumstances (i.e., time, funds, etc.) to get a puppy or rescue a dog.
I'm sorry if that means you have to wait, but I hope you can be selfless enough not think of what YOU want, but what is best for the dog who will have no say in the matter.