"In order to get along with a Basenji, you have to be at least half as smart as the dog!"
I think your first move should be to discuss this with your vet and have some tests run. Thyroid problems can cause behavioural issues. There are others on this board who have more experience in this area, but do a search on "thyroid" to see what has been noted in the past.
If it isn't a physical problem then you might want to consult a behaviourist. Dogs do change sometimes as they age, but also in response to incidents in their life. My boy became untrustworthy with strangers after we had visiting nurses coming and going frequently to deal with my husband's illness. I managed the situation by keeping him away from people he didn't know, particularly as they were leaving, because that was the danger zone for his aggression.
Until you find some answers I would restrict his access to anyone you think might trigger a reaction.
I think most are fine if properly socialized. Mine have varied from friendly to aloof, with my last one being somewhat untrustworthy and requiring supervision, particularly when people were leaving. Basenjis are curious and often will investigate purses, coats, etc. if given the chance. Theft of objects may occur.
Agree with Debra, you need to learn more about the breed. Behaviour with strangers is not a frequent complaint, but mischief and destructiveness are.....
One of the best strategies if you will be going out is to put the treats in the crate a short time before you leave and lock the dog out, so he can see them but not access them. Hopefully by the time you are ready to depart the dog will be anticipating getting into that crate and enjoying his treats! This can work very well and result in a dog that is looking forward to you leaving.
I did something similar with my dog's roller ball, loading it up and making him wait for it. This dog that formerly had separation anxiety would become impatient and sometimes baroo to tell me he wanted me to leave so that he could have his ball!
It sounds like you are making progress since she is settling down when you are out. That's good! Keep on doing what you're doing. If she is still restless at night you might want to bring her into the bedroom so she isn't alone. Many Basenji owners sleep with their dogs. Where I live winters are cold, and Basenjis like to be warm!
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.