"The only person since the Founding Fathers to draft more than one amendment to the United States Constitution," "the father of Title IX," co-author of the Bayh-Dole Act, etc.
http://en.wikipedia.org/wiki/Birch_Bayh However, as with so many people whose credentials look impressive on paper, it is not that easy to make him President.
The problem is that his only real chance was probably 1976, and his only hope then was to have the party's liberal wing (which I'll define as "everyone to the left of Carter") coalesce behind him. This did not happen; in the New Hampshire primary, he finished third, behind Carter and Udall, while fourth-place Fred Harris also undoubtedly took some votes that might otherwise have gone to Bayh. When he finished a disastrous seventh in the Massachusetts primary, his campaign was over.
https://en.wikipedia.org/wiki/Democratic_Party_presidential_primaries,_1976 If Udall hadn't run, would Bayh have had a chance--perhaps as a Midwesterner he can win the Wisconsin primary which Udall narrowly lost to Carter? (If Udall hadn't run, though, possibly still another liberal would have done so--maybe Frank Church would have started his campaign earlier.) Of course, even if Bayh got the Democratic nomination, it is questionable whether he would win in November. He would no doubt do worse than Carter in the South, and I don't know if he could make up for it in the North and West. (He could make Indiana competitive but was hardly guaranteed to win it.)
One thought: If Bayh had done well in the primaries, even if he hadn't defeated Carter, he might be chosen as Carter's running mate. But unless somehow President Carter meets with an accident or assassination or disease he didn't in OTL, this is not likely to lead to a Birch Bayh presidency--Bayh might get nominated in 1984, but I don't think his chances of actually winning are much better than Mondale's in OTL.
Another possibility: If Ford wins in 1976, Bayh could conceivably run for president again in 1980. If things get really bad during Ford's full term, the GOP brand may be so tarnished by 1980 that nobody, including Reagan, can save it, and any reasonable Democratic candidate, including Bayh, could win in November. But getting the nomination is still the problem. True, Carter's defeat in 1976 may discredit the "Democrats can only win with a southern moderate" line, and make it easier for a liberal to win the nomination. But again there is the problem that Bayh is going to face a lot of liberal competition--including very likely Teddy Kennedy this time.
(Another problem: Running in 1980 would probably mean giving up his Senate seat. AFAIK Indiana didn't have a law allowing someone to run for both Senate and president or vice-president the way Texas did to the benefit of both LBJ and Lloyd Bentsen. And with a Republican governor, Otis R. Bowen, the state is presumably not going to pass any laws for Bayh's benefit.)