States, however, should not back down from this issue about their right to insist on Constitutional adherence by the Federal Government. It's time.
As recently as 08 the state of California refused to allow a candidate on the ballot due to constitutional ineligibility.
Socialist party Candidate Roger Calero was removed from the ballot due to the fact that he isn’t a citizen of this country. Unfortunately he was allowed on the ballot in several other states.
It is up to the states to certify candidates since states run the elections according to their rules. They cannot however require a higher standard for national office than the US Constitution establishes. Regardless there is no constitutional control over a state’s authority to certify candidates as ITS responsibility.
As a result, each situation would become an individual case:
At the state level there would likely be judicial challenges to any refusal by a state’s election authority to certify a candidate to the ballot;
at the federal level there would be challenges to a state “overstepping its authority” by refusing to certify a (presumed) qualified candidate.
Either way it would not be good for a candidate with questionable qualifications.