Computing automaton for $L(A) / L(B)$ gives ones for $A,B$
This answers a different version of the question, in which the language in question is $L(A) \setminus L(B)$.
Here is an algorithm for deciding $L$:

Use the product construction to construct a DFA $C$ whose language is $L(A) \setminus L(B)$.
Let $D$ be a DFA whose language is $0^*1^*$.
Use the product construction again to construct a DFA $E$ whose language is $L(C) \Delta L(D)$.
Check (using BFS/DFS) whether some final state of $E$ is reachable from its initial state.
Output &quot;yes&quot; if no final state is reachable from the initial state. Otherwise output &quot;no&quot;.

As you can see, whether the languages encountered along the way are finite or infinite makes absolutely no difference for this algorithm.