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](http://matroidunion.org/?p=1106) for more information on finding shortest circuits in binary matroids.