Actually, it doesn’t say run. It says eligible. But, as per the state ELEC offices, the party putting the candidate on the ballot has to vet the candidate.
This is where Philip Berg’s suit has a standing. They didn’t respond to Berg’s suit either which was two parts. Fine, if Berg doesn’t have a standing on vetting Obama, so be it. But he, as a democrat, did have standing in challenging the DNC. The judge has no right throwing that out.
That is not what the Constitution says--the language is "shall not be eligible to the office"; a person who is not natural born is not eligible to act as President.
And running isn't injury--how is Berg or anyone else hurt by the fact that someone is running for President? Direct injury getting standing is where the person gets elected and can and does take action that damages your economic or personal interest or assets in some fashion. Even then, there is room for argument.
If you really want to try to understand operation of the standing issue and how someone might have standing, you should read the Michigan Law Review articles.