Unless you don't want to try to 'expand' the boards via adding humidity, then there's only one option left in my mind (w/o having to fill the gaps.)
IF you're fairly confident that the humidity level in the home won't fluxuate more than, say, 10% humidity, you should be OK to re-lay / re-install the floor boards. That's right - Remove the boards and re-lay it. And IF you do this, DO install some type of underlayment like 1/4" exterior grade plywood first. (Be sure to 'mark' the location of the joists so you can nail into them through the plywood.) I wouldn't install just a 'vapor barrier' because it will 'age' a lot quicker whilst exposed to the 'open air' (from the basement part of the barrier being exposed.) You may find out that you have a few inches of board space on the floor 'left over' to 'fill' when you re-lay the boards. You have a couple of options - Either 'borrow' a board from a closet or use a 'regular' piece of wood and somehow cover it up (or stain it to match as CLOSE as you can to the rest of the floor.) As long as the 'odd piece' is up against a wall, no one will really notice it (except you.)
Yes, if you hunt around hard enough, you will find something to fill in the gaps. But you really need to FULLY understand how wood expands and contracts and what it might do to whatever you put in between there. Mother Nature is a force to be reconed with. You will most likely end up with other problems AS WELL as the return of the cracks over time if you opt to fill. Have you considered area rugs and/or runners and/or mats??? You can cover as much of the floor as you want with these rugs and minimize what falls through the cracks. In time, the children will grow up and you can remove the rugs (or the kids - just joking.) It's hard for me to recommend something (like filling the cracks) if I won't do it myself. I do wish you well.