A minimum set hitting every base of a matroid

The problem is hard in general. Note that a minimal set that intersects every base of a matroid $M$ is a dependent set in the dual matroid $M^{*}$. Such sets are called cocircuits. So, you are looking for the shortest cocircuit of a matroid.

The shortest cocircuit problem (equivalently shortest circuit problem) is NP-complete in general (even for binary matroids). See this Matroid Union post for more information on finding shortest circuits in binary matroids.