We claim that the desired mininum is $\binom{n}{n-k+1}$. We see that for every element $E$ in $A$, $E$ appears in at least $n-k + 1$ subsets $A_i$ so that it is included in the union of any $k$ of them, but also in at most $n-k+1$ subsets, so that it is not always included in the union of any $k-1$ of them. We see that with this criterion met, every group of $k$ subsets has a union of $A$, so we only need to consider the statement that no group of $k-1$ subsets has a union of $A$. We note that if the elements of $A$ are distributed among the subsets so that no two elements always appear together, the Pigeonhole Principle implies at least one of the elements does not appear in the union of every $k-1$ subset if and only if $A$ has at least $\binom{n}{n-k+1}$ elements. It is always optimal to arrange the elements of $A$ this way, since not doing so would effectively reduce the cardinality of $A$ as two elements behave in the same way with respect to the subsets, so this value is optimal as well.