When my 16 year old Pomeranian became urinary inconinent, I had to make sure that I never acted angry, or showed any negative emotion, when she had an accident. She was very self concious of her problem and I did not want to make it worse. I could have kept her in my kitchen which has linoleum floors and would have been very easy to mop daily but she had a phobia about walking on that type of surface. I finally opted to put a large wire crate in my living room with an easy wash bed and lined with unprinted newspaper. This was used when I was not home and overnight. She had frequent urine "accidents" on the carpet but I just cleaned them up and then shampooed the carpet frequently. This was easily done with a 5 pound dog but would not neccesarily work with a Basenji.
I really like Lisa's idea of a litter box. A regular size cat box would be too small but one of those plastic underbed Rubbermaid boxes might be a good size. You want something large enough for the dog to not feel "cramped up". Just don't use clumping cat litter. If swallowed, that can kill and animal as the moisture inside the animal causes it to clump up and it won't pass. Another suggestion is "dog diapers". I would imagine many Basenjis would not tolerate wearing something like that but its an option to help keep the house clean.
My best wishes to you RockysWoman.