Yes they can keep people off ballots.
Courts have said SoS duties are ministerial.
The most recent case was Lindsay v. Bowen just decided in California. The Secretary of State does not have a duty to investigate a candidate but can remove an ineligible candidate is it is obvious or self admitted.
“Nor is this a case where a candidate’s qualifications were
disputed. Everyone agrees that Lindsay couldn’t hold the
office for which she was trying to run. Lindsay therefore
could never have been a legitimate contender for the
presidency, and there’s no doubt that “a State has an interest, if not a duty, to protect the integrity of its political processes from frivolous or fraudulent candidacies.” See Bullock v. Carter, 405 U.S. 134, 145 (1972). Holding that Secretary Bowen couldn’t exclude Lindsay from the ballot, despite her admission that she was underage, would mean that anyone, regardless of age, citizenship or any other constitutional ineligibility would be entitled to clutter and confuse our electoral ballot. Nothing in the First Amendment compels such an absurd result.”
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“While claiming that similarly situated candidates were treated differently than she was, Lindsay can’t identify a single person who appeared on the California ballot despite admitting that he wasn’t qualified.”
“Lindsay points to 2008 presidential candidate John McCain, who some considered to be ineligible to hold office
because he was born outside the United States. But, at worst, McCain’s eligibility was disputed. He never conceded that he was ineligible to serve, and it was generally assumed that he could. The Secretary does not violate the Equal Protection Clause by excluding from the ballot candidates who are indisputably ineligible to serve, while listing those with a colorable claim of eligibility. Because those two groups stand on a different footing, the Secretary is entitled to exclude the former while including the latter.”
http://cdn.ca9.uscourts.gov/datastore/opinions/2014/05/06/13-15085.pdf
There is also this for the 20th Amendment folks:
“But, even if it does, nothing in the Twentieth Amendment states or implies that Congress has the exclusive authority to pass on the eligibility of candidates for president. The amendment merely grants Congress the authority to determine how to proceed if neither the president elect nor the vice president elect is qualified to hold office, a problem for which there was previously no express solution.”