B's are a very unique dog.
I feel that basenji owners must have a great sense of humor to enjoy the quirks of these dogs.
I wouldn't have any other breed…but that is just me.
The more homework you do the better.
Visit some dog shows, and chat up some folks, then, check to see they do health testing on whatever breed you decide on.
Run screaming from anyone who says their breed has no health issues, I believe all breeds do!
Good luck.
We are here to help.
I agree that a really good place to start if you are still trying to figure out what breed you want is to make a list of what characteristics are most important to you. That will be really important when you are ready to start talking to breeders because there is also variation between individuals in a breed and knowing what you want can really help to match a puppy to a home.
So if you make a list and a basenji fits most of the criteria then probably if you are open with the breeder about what you want then they can work with you to help figure out if there is likely to be a basenji with those traits.
For me, I have no problem with people who are still in the research phase and not really certain if they really want a basenji visiting with my dogs. I think it is an important step and can really crystallize for someone whether this breed is something they want to continue to research or whether they should start looking at other breeds.
If your wife's biggest concern if the ability to alert and the issue of how much yodeling do they do, basic what is life with them really like type questions then I would say visiting with some basenjis would give her a better picture of what they are like in the home.
Just HAD to add…no shedding?? um not always!
When my two basenjis blow coat (at least 2x a year) the hair is just unbearable! I'm constantly vacuuming up little tiny B-hairs on the couch, rug, tables, counters etc. etc.
Good luck to you 
Next to basenjis, I love Rat Terriers. We have 2 siblings that hang with our basenjis and one of them is as big as my Eddie, one is tiny. They are short haired but do bark, are fairly obedient, usually nice dogs. I like the long-legged ones personally, and ther is a variety called Giant Rat Terriers that are the same size as a basenji.
Keep your mind open and you will find the right dog. Take your wife and kids to visit basenjis as well, you are getting lots of good advice from this forum!
My first basenji Wicket would hold my father in law at the door - which I thought was a good thing (he was a crabby man). I have NEVER not known that someone or something was in the territory or near or around my house. I've lived with basenjis since 1983.
Some are better than others with kids the better question is how are your kids with dogs?
If you are looking for something that is 'different' but sort of basenji-ish - there is a breeder of Wire Portuguese Podengo Pequeno's in the Northern Cal area. They do bark but this size not a lot - they are small 8-12 inches but hardy, they have a wire coat. Properly raised they are great family companions.
I have the big size in addition to basenjis but that would be WAAAAY to big if you are thinking of a Doxie You can click on my name and leave a message there - and I'll share some additional information.
Otherwise - find a dog show - go early and spend the day - looking at different breeds - you may find something unexpected that catches your eye.
I can't contribute much other than this: As everybody has said, find a reputable breeder. Visit them & ask them LOTS of uncomfortable questions. I'm sure somebody can post the "Important questions to ask your breeder" list. And you get what you pay for. If you find a basenji that has a price too good to be true compared to the others in your area, then it is too good to be true. You are paying for quality breeding, health testing, and a breeder who will be around for the rest of the dog's life to answer your questions, help you with problems or concerns, and, God forbid, take the dog back if you are no longer able to care for it.
Also, don't let your wife be persuaded or disuaded by what she reads about any breed. After a couple of weeks of researching basenjis online, I hated them! I was convinced they are devils that will destroy your house & ruin your life. I was only convinced when I saw a puppy, and who can resist a puppy of any breed? I will never own anything else!
As far as noise… our 2 year old male is usually very quiet, but grumbles when he's not getting his away. A baroo or yodel is a rare treat from him & we usually have to encourage him to do it. Our 1 year old girl never shuts up. She makes lots of happy noises to get attention, show her approval, whatever suits her. They're all different, and they're a great breed. They do require a good sense of humor.
Also, I found a great website last night that describes them quite well. It had me laughing hysterically. http://bleedingeyeballs.com/basenjiart/basenji.htm
lol @ hoodlums!!
"hoodlum" and "houdinis" is putting it mildly!
I think it greatly depends on the dog. I've read many differences here of Basenjis. I expected my B to be destructive, noisy, not wanting to go outside while raining/snowing, etc. But I found the exact opposite.
He only makes noises briefly at times. Usually when he wants attention or when he's nervous (in the car). It's a yawn with a quick howl/baroo at the end. Hard to explain but it's the cutest thing. Other than that he's 100% silent. Not all B's are alike from how i've gathered things. I found my B to have 0 destructive habits. The breeder actually gave me a heads up on this before I picked him up originally, and it was true. I can leave him at home while i'm gone during the day and he doesn't move a single thing. Of course we know not all B's are like that.
Just do your research and understand that it all depends on the dog, but use the information about the breed as a base.
My 2 are polar opposites in the house. Ty is a laid back couch potato that lives to be petted. Katie is a gung-ho, pester you to the end of the earth, come play with me or I'll destroy something, kind of gal.
They both shed. And by shed I mean as soon as the Vernal Equinox hits they blow coat. I vacuum up what seems to be the equivalent of a 5 dog household worth of hair on a daily basis (and yes, I do brush on a daily basis–that's just what's left on the floor and furniture in between brushings)
Guard dogs? Not in the typical sense. I don't have the bark differentiation that you might get with other breeds; friendly bark for a welcomed guest, alert bark for a stranger, notifying bark for the postman, etc.. I get a "get off my front porch" scream with the back burred up, or the "back off my fence a$$'ole" snarl with teeth bared and back burred up, or perhaps the quiet grumbling tail wagging with the back burred up "welcome friend" signal followed by the big baroo!! (The "back up" on this breed (IMHO) signals to strangers as aggression. I found my 2 use it to signal excitement–they even burr up when chasing small critters in the back yard!!)
From dog to dog, my 'sengis are VERY individual in personalities.
The discussion between nature vs nurture should never be discussed when it comes to this breed. It would be a moot point. Even Darwin would be confused and probably throw his note books in the air in frustration and sail away to Key West to spend some 'normal' with Hemingway:D
And don't be fooled when someone says "oh he's a little angel". EL D came to me with a reputation as being very destructive but never did any damage once I got him home and my current Luny had a reputation for being very nice in the house and turns out he has a severe anxiety problem and will destroy anything by the door if I leave with him loose.
So be sure to watch the pup/dog for awhile to see how s/he acts on its own and with other dogs and also pay attention to how a dog reacts to you, a stranger.
