Prmarily, it's the responsibility of the party organization which nominates its candidate. That's why the DNC is a defendant in Berg's suit.
If the party organization attempts to place a constitutionally ineligible candidate on the ballot in each state, the view of many is that the secretaries of state of the various states (or whatever state official or state agency has authority to conduct elections) have responsibility to correct the error of the party organization. Remember that a presidential election is actually 51 separate elections - one in each state and one in DC.
I thought it was 57 :)
Seriously, thanks for the info.