The problem is hard (not in P) in general.  Note that a 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).