Then who does have the responsibility to know if the proposed POTUS has the qualifications to be President before He/She is put on the ballot?
Since there is no federal election for president, the task of determining eligibility falls on the states, probably to the secretary of state of each state.
Each state sends members of the electoral college, each state runs its electionin its own way, so each state has to verify that the candidate is OK.
There were states that allowed known, proven non-citizens on the ballot (some Mexican radical forget the name sorry), which is a small hint that nobody actually does check regardless of what their responsibility is. Probably because most peoploe think there is a federal election (and we live in a democracy).
So it would take just one or two states to make the others look foolish? (...at least I think in this case)