Makes absolutely no sense to me. A healthy betta, or any fish for that matter, can occupy any level in the aquarium it chooses from bottom to near the surface. Passes my understanding how water depth could be a problem.
With that said,, I have heard that those who breed corys usually keep the babies in shallower water so that it is easier for them to rise to the surface as needed. Perhaps it is similar for young Bettas. As adults however, I can't see how depth would have any measureable negative effect on the health of the fish of either species. JMO