I would look on Petfinder and search under Basenji, and even Hound and/or Terrier and put in your city. You can narrow your search further by age, gender, size, etc.
Jennifer
Hmm, that's strange that nobody responded to your e-mails. Sometimes it takes a couple days for people to get back, but most BRAT people are very prompt.
I would fill out a general BRAT adoption application anyway, and indicate that you are interested only in mixes. I receive a lot of notices from about mixes in shelters and looking for homes via the BRAT adoptions mailing list (I'm not looking for one at the moment, but I was and still like to be kept in the loop – you never know!). Of course, it's a national mailing list so being on the West coast, it sometimes drives me batty when a perfect-sounding guy is on the opposite side of the country. But if you are truly committed and if the paperwork pans out, there are ways to transport dogs from nearby states.
I checked out BRAT today, and for basenji mixes, I believe it says that you contact the owner directly and that BRAT is not involved. Which confused me, because after all, they are being listed on the BRAT website.
I think it's a matter of utilizing the BRAT internet network. The group's mission (and funds) are designated for purebred rescue, but naturally there are many good people out there who would be willing to open their hearts to a mix, too. So BRAT's site will list them out of courtesy, and people on the mailing list will receive notices about mixes if they express interest (I'm actually not sure if it's a separate list or not – I get all notices for available adoptions, so I haven't paid much attention to the sender). But BRAT funds are not earmarked for fostering, providing vetting services, etc. for mixes, so they're not "officially" part of the BRAT system.
Hopefully Joanne (giza1) will pop in here to clarify, as I believe she handles all the BRAT Mix listings.
BRAT doesn't handle the mixes, they simply allow them to be listed there. BRAT is overwhelmed with purebreds. In the past I contacted 3 different listings, 2 got back fast, one I never heard from. Please don't blame BRAT. But why not look at neighboring states too. And Petfinders. 
BRAT has a good number of basenji mixes on its website: www.basenjirescue.org. Hit the link on the home page: View Baseni Mixes.
BRAT's basenji mix coordinator recently signed on a co-coordinator, so you may get more of a response. Otherwise, try Craigslist. I see MANY on there I search CL for BRAT for any Bs needing to be rehomed).
There is a male mix in the AlvinTx dog pound, I think he is on petfinders, this is from a friend of mine. Good luck finding your B mix.
Here is a basenji mix that is on Craig's List… perhaps you can be this boy's hero?
http://dallas.craigslist.org/dal/pet/2182267984.html
Date: 2011-01-28, 7:22AM CST
Reply to: comm-dmvsd-2182267984@craigslist.org
We have a small dog that needs a new home. He is a very loveable, submissive good dog and want to find him a good home that will give him attention. We have a 2 year old baby and another one on the way and have been spending less and less time with him.
Some details:
I am not asking for a rehoming fee and I will supply you with a nice doghouse, 2 dogbeds, collars, leashes, toys and dog bowls.
If you are interested and would like more pictures please let me know. He is really a good dog and I want to make sure he goes to the right person.
I hate people.
