How does the DNC decide who they will let run as a Democrat? What would happen if a candidate applied to run for president as a Democrat who strays very far from the platform? Would he be denied, and on what grounds? Can a candidate be denied for being to far left, too far right or in some other way in conflict with the core values of the platform?
The rules for being on the primary ballot vary from state to state. In South Carolina, the party reserves the right to deny ballot access to a candidate who has been a member of another party or may show disloyalty to the party. In New Hampshire, only a filing fee is sufficient to get on the ballot.
As for getting the party's nomination, this is decided by hundreds of delegates at the Democratic National Convention. A candidate needs to get at least 15% of the vote in a state in order to get any pledged delegates from that state. A majority of delegates are needed to win the nomination.
It's a democratic process, if you want to run a far right campaign and you can get the votes in the democratic party you can become the nominee. To be in the debates you have to poll at a certain percentage and you have to have a certain number of donors. If a right wing politician campaigned and won in the democratic party it would be reflective of a change of values of the party or it would be reflective of an attempt by the opposition party to infiltrate the party, neither side wants to do that because it would set a precedent that would hurt them both.