@Mowgli:
We also have heard that sometimes dogs with anxiety agitate the other dog causing both to act up when left alone?
That is certainly possible. However, my first male had separation anxiety/confinement anxiety, and when we first got him we did crate him, as we weren't comfortable leaving him loose with our girl. We did not crate her, but did confine her to the room he was in. She ignored his antics, but he persistently would rock & roll the crate from one side of the room to another, and destroy anything he could get at. We put that crate inside a larger one for a bit, than worked at making him safe in the house. (we did a lot of role playing…...getting ready to leave, leaving and coming right back or returning via the other door, etc.) In relatively short order he could be trusted to do minimal damage, and we started leaving him loose too.
Have you considered a web cam or baby monitor video cam so that you can see what happens when you leave? Sometimes that can be quite revealing. My guy would invariably do his evil deed immediately after we left, then settle down and be fine until we returned (unless we ran too close to his meal time.....he would get restless if he was getting hungry and his "staff" hadn't gotten back to make his dinner!:D )
He never destroyed anything important except for one of my husband's shoes that he totally ate.....except for the sole! Well, he and I had a serious discussion about that shoe, and that was the end of his destructive ways except for a token piece of paper he would rip and leave on the couch for me to find. I would verbally chastise him for the rippage. My husband said, "why bother" and I said, because if I don't "notice" it he will escalate.....and I am sure he would have. He was giving me the finger for going out, and needed me to acknowledge it. I know people who say don't attribute human motivations to dogs, but I figure they don't know Basenjis.
By the way, I think it is absolutely key to not make a big deal of coming and going when you have this problem. Our current dog seldom even gets off the couch when we come home, and even if he does I do not immediately return his greeting. I keep things absolutely low key.