This is a naive answer.
The intersection over an empty collection of sets does not make sense (to me). The reason is this:
Suppose $S = \emptyset$. Now let $x$ be any comprehensible object (set, element whatever). Now I ask you why is $x$ not in $\bigcap S$. The only reason for $x$ to not be in $ \bigcap S $ is if $ (\forall A \in S, x \in A) $ is false. For this formula to be false there must exist a set $A'\in S$ such that $ x \not \in A' $ i.e. the negation of the "set-builder" formula is $ (\exists A' \in S \in , x \not \in A')$. But there can be no such $A'$ in $S$ since $S$ is empty.
Hence, anything belongs to $\bigcap S$. Whether this is a problem is for you to decide for yourself. It depends on the level of set-theoretic formlisation you wish to incorporate into your mathematics. If you believe in a "Universal Set" then no problem! But most would have seen big fat issues like Russell's Paradox resulting due to the thought of inclusive sets.