Qualifications for the Office of President
Age and Citizenship requirements - US Constitution, Article II, Section 1
No person except a natural born citizen, or a citizen of the United States, at the time of the adoption of this Constitution, shall be eligible to the office of President; neither shall any person be eligible to that office who shall not have attained to the age of thirty-five years, and been fourteen years a resident within the United States.
Sure they are. But we aren't at that stage yet. We won't be until the electoral college votes (see Amendment 12). And with our system, that is after the general election. Again, if the Dems put up someone who is not qualified, those electoral votes will be thrown out, and, in our two party system, the Republican will end up with the most electoral votes and win the presidency.
So right now, it is up to the Dems to choose wisely. If they don't. Too bad, so sad. They will get in the history books and the worse DNC in history.