It's hard to gauge "public" suspicion without doing a deep-dive into journals, diaries, etc., but amongst sports writers and even former players, there was definitely talk of something fishy during the series. Hugh Fullerton was probably the most vocal and prominent of these voices, as he'd heard rumors before the Series even started that the Reds were "a lock" despite the White Sox being the far superior team on-paper. This put him on the scent, and throughout the series and after, his articles for the Chicago Herald-Examiner began the process of blowing the whole thing open by singling out specific players and plays that didn't pass the smell test.

Fullerton wasn't alone, either. He watched the Series with former Giants pitching legend Christy Matthewson, who also agreed that the Sox-Reds Series was not on the level. Fellow sports writer Ring Lardner also voiced suspicions during the series (and allegedly trolled the team with a song during a train ride, though this story, told by Daniel Okrent in Ken Burns' 'Baseball' doc may be apocryphal). So people were indeed talking and writing about it while the Series was going on, yes, and some of these folks were voices with some reach.

What we do know is that suspicion of a fix was strong enough that the White Sox owner, Comiskey, ordered his own private investigation into the matter following the conclusion of the Series. So while it wasn't conclusively known that the Series had been fixed when it was still being played, enough people knew about the suspicions that only somebody with their head completely in the sand would have been clueless about the rumors.