Are there criteria that will make: $A \subseteq B$, $A$ unrecognizable imply $B$ unrecognizable?
The condition about $B$ can be transformed into a condition about $B\setminus A$, as follows.
If $B\setminus A$ is decidable, then $B$ is unrecognizable.
Indeed, if you could recognize $B$, you could determine whether a word $w$ is in $B\setminus A$. If it is, then it's not in $A$, otherwise it is. So $A$ would be recognizable as well.

Note, however, that the converse does not work - there are sets $C,D$ such that $C\cap D=\emptyset$, both $C$ and $D$ are not recognizable, and $C\cup D$ is unrecognizable as well.