If your not going to show or breed your basenji, then why not give the dog some peace of the hormone surge and fix him?
Again, there are pros and cons. I've done my fair share of research on this topic. He has never tried to mount anything, doesn't mark territory, etc.
There hasn't been a behavioral/health reason to neuter him. For his own well being there hasn't been a convincing reason to go with neutering. He's a happy and healthy dog. These anxiety issues are minimal compared to other dogs. He eats well, sleeps well, and is one of the best dogs I've ever had.