New Hampshire election law says anyone can be on a presidential primary ballot who pays a fee of $1,000 and fills out a Declaration of Candidacy. However, the Declaration of Candidacy since 1983 has said, “I am a registered member of the (fill in the blank) party.”
The literal language of the New Hampshire requirement that the candidate be a registered member of the party has never been applied, because 20 states don’t have registration by party. Walter Mondale, the Democratic nominee in 1984, was not a registered Democrat because Minnesota has never had registration by party (and, if it did, Mondale would have been a member of the Democratic-Farmer-Labor Party). Bill Clinton was not a registered Democrat for the same reasons; neither was Al Gore. Among Republicans, neither George H. W. Bush nor George W. Bush was a registered Republican because Texas doesn’t have registration by party.
So, in practice, the affidavit doesn’t mean what it says; what it does mean is that the candidate must hold himself out as a member of the party. See this story. It will be interesting to see how Senator Bernie Sanders handles this situation. Even though he has announced that he intends to run in Democratic presidential primaries, he has not said that he a Democrat, and he lists himself in the Congressional Directory as an independent. Thanks to Rick Hasen for the link.