Minimum number of edges to remove to disconnect two node sets $A$ and $B$ in a directed graph
You could try reducing your problem to the directed $s$-$t$ cut problem. 

That is, given an instance $G=(V,A)$ of your problem, construct a graph $H$ that is initially a copy of $G$. Then, add to $H$ two new vertices $x$ and $y$, add an arc from $x$ to each $a \in A$ and from $y$ to each $b \in B$; all with a large weight (say $m+1$, where $m = |A|$). The intuition is that an $s$-$t$ cut of $H$ will never use any of the arcs originating from $x$ or $y$ since they are so heavy, but instead only use arcs present in $G$.

You should try to formalize this intuition, i.e., (i) show that the reduction is correct and (ii) show that there is an efficient algorithm for directed $s$-$t$ cut. If both check out, you are done.