Certainly that should be the responsibility of EACH and EVERY state Sec of State before allowing a candidate's name on the ballot. Otherwise, many of the voters in that state, who vote for an unqualified candidate, risk being disenfranchised.
I asked the same question in Georgia and was told that 'the Democratic Party Convention certified 0bama's eligibility and that was accepted'. To me, that is totally unacceptable and demonstrates that the Sec of State's office has abrogated it's responsibility, not to mention the propensity of the dems to lie.
What they should be required to do and what they are required to do is the gist of McInnish v Chapman.