24" would be optimal if you have the materials for it. If cost/material is a factor, I feel that 16" is about as short as you can go and still have some decent level of protection.
I believe (and I could be wrong) that weasels aren't much for digging, however they will follow rodents through their burrows, so the aim should be excluding rats and other burrowing rodents.
A 12" apron may or may not be wide enough to exclude a rat from getting in in 1 night, as I've seen a...