There was nothing wrong with White's 9.h3, the game was lost on some poor moves thereafter.
Taking advantage of a h3 or ...h6 move for checkmate!

Thank you, Hastonian. Can you please diagram how White could have held on to win against Black with proficient play from both sides after 9. h3 in that position?
And what moves of White were suspect? Keep in mind I'm not trying to bother you, the .pgn is just something I read in Chernev's Logical Chess, with the annotations being my summaries of Chernev's commentary.
Thanks a lot,
Cheers, Becky

Well, 10. dxe5 looks like a terrible mistake to start with. Why resolve the tension in the center and activate the black pieces unnecesarily. After that, 11. Nxe5 can't be good (although I don't see much better), but white could still probably survive with something like Kh1 (although black then wins e4). After dxe5 the game goes from a roughly balanced position (white might have a slight edge, but it's hard to develop his b1 knight) to an extremely easy game for black where he is probably just winning.
Ever play an opponent who, in a Kingside castled position, insists on playing a pawn up h2 to h3, or h7 to h6, trying to forestall a pin on this f3/f6 Knight?
Here is an instructive game I read in Logical Chess move by move (1957) that shows how a master takes advantage of the weakened kingside pawns and demolishes the opposition, Gangnam style!
As Santasiere said, "Touch the pawns before your King with only infinite delicacy."
Cheers, Becky