This a tough one. My thought/vote is that it would go to Babe Ruth. I never saw Babe Ruth live (and he is not of my era), but if you can consider that it's hard to determine outside of a particular era, we can only make some best-guesses.
1. As a pitcher, he was a 20-game-winner as well as World Series winning pitcher.
2. Of his era (and beyond), he held the HR total record for serveral decades, until Hank Aaron broke it in 1974.
3. He was one of the main contributors to making baeball the nation's pastime with drawing massive amounts of people into the park. It sure as hell wasn't Ty Cobb that people went running to go see.
However, if how you define best MLB player as having the most all-around talent (and records to go with it) in all the skills (minus pitching), I'd say Willie Mays is the leading candidate.
Joe DiMaggio is not far behind.