Exercise is the best way without medicating.
You should also work on crate training. Does your dog freak out just getting into the crate? Ours used to pee as soon as he got in it- like he just freaked out and pee'd everywhere. We had to work on him. And he still howls like crazy.
Some people have had success with an herbal human remedy called Rescue Remedy. I got mine in the organic/herbal section of a fancy grocery store. It did absolutely nothing for our dog, but some apparently seem to do well on it. It's supposed to calm them down.
If all else fails, you should consider a trip to the vet to discuss stronger meds to help with his anxiety.
But of course, the best solution, IMO, is to just get another Basenji. Like others have said- they just want to be together.