Short answer- NO!!!!!!!!!!!!!!!!
Basenjis are hunters and most will do anything in their power to get to the critters in their environment. Mine spend most of their outdoor time looking for chipmunks that live under the shed. If you decided to get the hamster it will have to be kept in a room where the basenji can't go. I have and do keep basenjis and small animals in the same house but they aren't allowed near them. I don't think they even know I have the geckos and snake!
Some dogs are fine with small critters but I personally would not let them near each other. It's not worth the risk in my opinion. Accidents happen in the blink of an eye!
Also, rats make much better pets for kids than hamsters. ;)