Most Democrats stood by him, including most prominently LBJ and Ted Kennedy. Just most Democrats song McGovern was crashing and burning and didn't go out of their way to beat the drum too loudly.
LBJ support for McGovern was, if anything, only a public display and likely not genuine. Richard Nixon in his autobiography wrote about a telephone conversation with LBJ in which LBJ said that he would support the Democratic ticket at all levels because of his stand within the party. But he would not stop any of his colleagues to support Nixon (like John Connally and his press secretary George Christian). And he provided Nixon regular advice during the 1972 campaign.
I suspect that in the end, LBJ voted for Nixon--after all, McGovern was pretty much LBJs enemy since the escalation of the Vietnam conflict. And it really didn't matter--Nixon crushed McGovern in Texas.