The short answer I believe is don't worry about it.
Technically yeast can become airborne but the resulting culture is very small, bordering on insignificant in the face of a different overwhelming strain of yeast. But as long as all the vessels are active, and out gassing from their own fermentation, there is really no path in ingress of one of the other yeasts.
I would be more concerned about brewer induced cross contamination. Not having your utensils sanitized between each and every use. Spoons and hands both. And if it were me, I would stir each batch in a different location from where they are placed. i.e. fermenting in the fridge, stir on the porch/garage/far side of kitchen, etc.
Finally, noting you are finishing in "the fridge", I assume you mean your everyday one that unless you are one of the 0.01%, is a microbiology culture fest (no offense meant, just reality). Assuming your sake has not been contaminated in there, that there is pretty good circumstantial evidence that other cultures are not making it into your batches.