What you are describing is separation anxiety and it can be tough to deal with. Personally I do not agree with isolating a Basenji at night. They like to be with their people and will generally settle down and sleep quietly if they are close to you. Especially if you will be leaving them during the day, I think it is important not to leave them all alone at night.
Developing a strategy for leaving the dog during the day may require some experimentation. Some dogs do not like crate confinement and do better if kept in a "dog proof" room where they have more space and can perhaps look out the window. Many become trustworthy enough to leave loose in the house, but you don't want to chance this until you have more experience with your dog and she is older. Whether in a crate or not, it's important to give her something to amuse herself, perhaps a stuffed Kong for her to work at, or a meaty bone if you feed them, or maybe a puzzle game of some kind where she has to work to obtain a food reward. I used a roller ball filled with treats for my boy and he looked forward to my departure because he knew he would get his ball.
You may have to start over with making very short departures and returns, gradually leaving her for longer. Make leaving and returning as matter of fact as possible. Don't make a big fuss over her when you come back. Ignore her if she fusses over you, and go about your business for a few minutes before acknowledging her.
If you have access to a video device, it can be very useful in determining how upset she actually is when you leave, as many dogs are destructive or lose control of their bowels right on your departure and then settle down. If she stresses the whole time you are out, you need to figure out a way to prevent this, perhaps doggy daycare if it is available.