Ok, I think I understand the question now. To clarify, the free profinite group of rank $n$ may be regarded the profinite completion $\hat{K}_n$ of $K_n=\mathbb{Z}^{\ast n}$ the free group of rank $n$. An open subgroup $H< F_n$ must be finite index since $F_n$ is compact, and the cosets of $H$ cover $F_n$. Thus, $H$ induces a finite-index subgroup $J< K_n$. Then $J$ is a free group of rank $$rank(J)=1+[K_n:J](n-1)=1+[F_n:H](n-1).$$ Since $J$ is finite-index in $K_n$, $H=\hat{J}$, the profinite completion of $J$, and therefore $H$ is a free group of rank $rank(H)=k=1+[F_n:H](n-1)$. Thus, your question is equivalent to asking for the index of the intersection of all subgroups of $K_n$ of index $\leq \frac{k-1}{n-1}$. I'm not sure this has been worked out, except possibly for $k=2n-1$ corresponding to index $2$ subgroups, where of course the answer is index $2^n$. One may obtain a crude upper bound by counting homomorphisms of $K_n$ to $S_j$ (where $j=\frac{k-1}{n-1}$), which is at most $j!^n$. So one has a bound on the index of the intersection of stabilizers of all of these homomorphisms of $j^{j!^n}$ :).