Let us suppose we have a finite ground set $E$ and a collection of sets $S_1, \ldots, S_m$ which we can assume without losing much generality is closed under intersection and we can similarly assume that $S_1=E$.
There are matroids on the set $E$ such that all of these sets are closed (in particular, in the free matroid every set is closed). As such there is such a matroid of minimal rank.
As a lower bound, if there are $k+1$ of the sets forming a chain, then since closed sets are maximal for their rank, they all have different rank, so the rank of $E$ is at least $k$.
Is this lower bound ever not attained, ie is there some intersection-closed family $F$ of subsets of a ground set $E$ (with $E\in F$) with no chain of length $k+2$, but such that any matroid on $E$ for which all subsets in $F$ are closed has rank at least $k+1$?