It is wonderful of you to want to give this pup a good home. First, you need to determine if she has a responsible owner who is looking for her. Next, you need to find out if she is a purebred or not. If she is a mix, she might be okay living in your yard. All of the many basenjis and b-mixes I have known hated to be outside in the dark at night, even with a human! It's a weird quirk of the breed.
For just about any other breed, your situation would be fabulous. For a purebred basenji it is not! I agree with everything everyone here has said. Basenjis need to live with people. I can guarantee that in time, she will let you know in no uncertain terms how unhappy she is - whether by digging to China in your yard, escaping, damage to your property (like digging a hole in the door to get inside), etc. It will take awhile, but it will happen and it will be very annoying.
If you don't want a dog to live in your house, but you still want a dog, you should look for a breed that will be happy outside. It would be so much kinder for the b-pup to go to a home that will let her be inside. It is possible that her previous home expected her to live outside and that is why she was running loose.
Please do not give her to the local animal shelter - they won't find her the right home. If she is not a purebred basenji, Basenji Rescue will still try to help, but they won't be able to take her into foster care. Good luck!