All Sanders' supporters need to do is convince Clinton voters that they should stay home.
They won't do that because they know Sanders still has a chance and will until June 7th (more than likely). If he has even a 1% chance (obviously a conservative estimate) then it's not mathematically over. Clinton supporters know this and are going to vote, but to argue that Sanders has a zero percent chance is to argue that Clinton supporters should stay home, which they are not doing and therefore are not being 100% honest.
If Clinton wants to win in November she can stop attacking Sanders any time, but she is not doing this. If Sanders attacks Clinton, it's not necessarily personal, it's (unfortunately) seen as the way to win. It might be ok for him to focus more on a positive message, of course. So I have no problem if they kiss and make up. You can blame both sides, but if you blame only one side, you are not sincere in wishing for a united party. Although I have said that I think Sanders is being premature if not unrealistic to be focusing on super delegates who are not likely to switch right now. He should wait until after June 7.
He shouldn't do anything after June 7 except drop out because trying to win a nomination when someone else has already won a a majority of pledged delegates is a crappy thing to do.