Len has the right of it to some extent but you need to find some kind of group where the trainer has experience of Basenjis. Remember, they are not as other dogs in the way they react to training. General socialising is probably better than formal teaching in a group, although this is a good idea too, so long as the trainer knows and understands Basenjis.
It occurs to me that in the cases of nipping, biting and even leaving marks, the Basenji has been allowed to become the Alpha - to get the upper hand. This is never a good idea !
It could be they just don't like the person they are biting or it could be attention seeking.
Somehow you have to get through to them that this is not acceptable behaviour. A sharp, and immediate, tap on the muzzle with a firm 'NO !' Use voice to express your displeasure (just as you do to praise them when they are good). And having told them 'NO' - that is it - end of game, end of attention. You turn away from them and let them learn that this sort of behaviour turns you right off them. In the end, they will learn. Withdrawal of your attention following each biting episode and a a vocal admonishment should finally get through to them that you will not tolerate it.
Aggression towards other dogs can be 'time of year' - again, you need a trainer who knows Basenjis.
I have never experienced any form of biting in my pack. But then, with a pack it is more than ever important to establish ME as the Alpha. However, you should try to gain that status !
Good luck !