The statement that for $X$ compact Hausdorff that is ccc (there is no uncountable family of non-empty pairwise disjoint open sets) every intersection of $<2^{\aleph_0}$ open dense sets is dense is equivalent to Martin's axiom MA.
This is independent of the usual ZFC axioms of set theory, and is a very well-studied axiom (especially its consequences in topology and measure theory).
Without the ccc condition on $X$, the statement is false in ZFC. For compact metrisable spaces ccc follows (as these are separable, and this implies ccc), so it would be a consequence of MA. I don't think it is a ZFC fact, though I don't have a consistent counterexample at hand.