I found that my little Cairo (6months) would need to go to bed at 9pm every night and Caesar (6yrs) would stay up for a few hours alone with me. Cairo's kennel is in the living room so he could see us and would cry and cry….
I put a comforter over the kennel and he quieted down immediately. He must have related it to a den. Try putting a blanket on the kennel when it is time to go to bed to establish it is quiet time and give your basenji a nylabone or treat that takes a while to eat. It should distract him a bit until he settles past that tantrum point.
I use my kennel for a time out spot as well and learned early on that you shouldnt acknowledge the bad behavior by taking them out when they are crying. They need to learn that they must be quiet before let out of the kennel or they will train you to let them out.
If you want to train him into the kennel it is only fair that you have a schedule so he knows he is going in because of the schedule, not for punishment.
It sounds like he is your baby which is why he cries to you. Just establish a schedule for him and he will get used to it.
If he is screaming like he is going to die, well that may mean you need to try something else, this is just for the LET ME OUT cries.
These are just my experiences. I hope they help a bit. Keep us posted!