Baseball games are definitely family oriented: cool and relaxed atmosphere, lots of young kids, lots of songs [clap]: check out on the web the lyrics of "take me out to the ball game"[msnwink]
Of course you could be sitting next to the ultimate fan who'll get excited and throw peanuts at the players, but there's little chance of that (did happen to me once though: had never met such a perfect image of a redneck before !
I can't imagine a baby would be a problem, unless he/she starts screaming at the top of his/her lungs and can't be stopped, which always seems to be the case in the most embarrassing situations (deja vu...[msnembarrased]).
The game is really easy to follow but can be [u]very long</u>, so you want to sit in the shade if you're going day-time specially with kids. When I was living in Kansas City we used to bring a cooler and have a family picnic there... not sure they allow that anymore[msnscared]? There's always plenty of (junk) food anyway: huge bretzels and hot-dogs being my personal favourites. With a Bud of course[beer]