Jane
I have an 11 month old female and noticed a difference in dog parks after her first season at Christmas - she's become a little selective towards
other dogs and intolerant of annoying behaviour, at times getting aggressive in response to it
If her recall was better I would still take her and call her back at the first hint of trouble, but she won't come back when called if distracted (despite lots of practice) so we have stopped going.
She goes out for walks on the lead a couple of times a day and greets other dogs, occasionally getting a chance to play when on the lead and we hire a private dog walking field weekly for off lead exercise and recall practice. We also have a big garden.
It would be nice to have been able to continue with the dog parks, as when it's going well she gets so much out of it but I don't want to be responsible for an injury when I can't reliably call her back yet.
@jengosmonkey haha, here's a subsequent version 
https://www.instagram.com/p/COXYePshG4F/?igshid=1qyldm5djqkcx
Right sided photo to follow - for some reason she always seems to pose with her left side showing!
@JBuckee if you do consult with a behaviourist, it can be helpful to try and find one with experience in working with primitive breeds.
We also have an 11 month old and I have had to spend a lot of time training my children to recognise when she doesn't want affection e.g. lip smacking, yawning and looking away. With consistent reminding they now understand and sometimes have to remind me They are older though (8 and 14) so easier to train.
As already suggested a separate Basenji sleeping area that is out of bounds to your 3 year old would be helpful. We use an open crate with a blanket over the top and none of us approach it when she does in there for peace and quiet.
How old is your other child? Can they help to remind your 3 year old?