Guess what, grooming a Basenjis is VERY easy! An occasional bath and weekly nail and teeth attention does the trick. Here are some tips.
Nails should be trimmed, ideally ground using a nail grinder, once per week. This keeps the quick (the blood supply) short. I recommend a Dremel tool with the sandpaper setting. You can buy a grinder specific for dogs - I personally don't like the guard as I can't see as well. My Dremel is a home tool purchased from Amazon. If you have multiple dogs, I recommend the diamond grinder attachment. The diamond grinders are pricey, but they don't get hot like the sandpaper attachment.
I brush teeth 1-2 times weekly with a baby, soft toothbrush and dog toothpaste. My dogs love the taste of the paste and love getting their teeth brushed. Alternatively, you can do the dog toothpaste on gauze. If you keep up regularly brushing, you can potentially go years without an expensive vet teeth clean.
I bathe infrequently. Basenjis are so generally clean, regular bathing is unnecessary. In the summertime, our yard gets dusty so we bathe 1-2 per month, but other seasons baths are only as needed. I bathe in my tub at home with a rubber, non slip mat on the tub floor. Spending $15 on the mat is important - dogs are so much happier being bathed when they aren't worried about slipping. You can bathe with dog shampoo, and I think Dove bars work great too!