The POD has to be a stronger LBJ, there's no other reason why RFK wouldn't run. I'm not entirely sure RFK wanted to be either a Senator or President, I do believe if JFK had lived he would have been happy to never bother to run for office. That said, with JFK dead RFK running for President is inevitable (IMO)…*I can buy you four years, and perhaps RFK loses in '72, but with everything I know—Bobby was going to have to run for President, whether he wanted to or not. (I'm not even certain RFK would stay in the Senate, actually.)
Anyway. Let's say Viet Nam goes slightly better (Tet offence is portrayed as it really is: a crushing defeat for the North Vietnamese) and LBJ staves off the McCarthy challenge.
1968 sees Nixon lose to LBJ, 1972 has Reagan beating RFK.
So RFK has lost, but retains stature equal to or better than the actual President. Assuming he stays in the Senate my bet is that he serves to define the future of the Democratic Party. Without McGovern the Dems retain equal footing on national security and with RFK they can outline a free market pro-poor/black/urban agenda without requiring big government programs (though those too will be a part).
Paired with the somewhat more pro-government Ted I imagine they define the two wings of the Democratic Party for the next half century. With RFK's '72 run and lose Ted Kennedy will not feel the same pressure to run; with both EMK and RFK alive they will exert a very strong influence even if neither become President (and, to be fair, either could at some point in the 70s or 80s).
The various specific issues are less important: a Democratic Party able to articulate a non-Republican ideology (I'd argue Clinton defined Rockefeller Republicanism quite nicely, but wasn't much of a Democrat) and present an actual alternative to the Reagan conservative ideology.
At the very least RFK's survival ensures LBJ will have to put forward a new cities agenda, it ensures that Reagan will not fully capture blue collar Democrats, and it means that (if he so chooses) there will exist an almost semi-presidential system (see France, Russia) in the United States—whoever the President might be, he'll probably need practically Prime Minister Bobby to go along.