The core of his supporters will vote for Hillary as they understand that the alternative is unfathomable. That does not mean all of his supporters will support her or that it will be enough to beat Trump. There will be a divide among his supporters of those who will support her and those who will not.
Bernie Sanders' key constituency is likely to vote for Hillary Clinton in the general election. There are two major parties, and most will realize voting for a third party will tip the election to Trump. Sanders has endorsed Clinton and appears likely to stump for her, which should help ease ill feelings.
The vast majority of Bernie Sanders supporters have already decided to vote for Clinton. However, those who remain in the Sanders camp are becoming increasingly more vocal about their opposition. This is similar to 2008 when Clinton supporters formed PUMA. By the time the election rolls around, these people won't have any significant effect.
The movement "Bernie or Bust" alleges that Bernie Sanders supporters will vote for Donald Trump if Hillary Clinton wins the Democratic nomination. This is only bluster: Sanders supporters will eventually see the error of their ways and realize that having Clinton in the White House is a lot better than having Trump in the White House.