Out of Many is the textbook that I have used most often in my teaching US History at the college level. In my opinion, the text argues that the organized sports did more to bring people of different classes together. However it does talk about how the National League (baseball) was more for middle class people (no alcohol sold, no games on Sunday, higher prices) than the American Association. So that is both reflecting and shaping social divisions. Having sports become a big deal also leaves lots of women out and deepens division between the sexes. (End of Ch. 19 -- "National Pastimes")
The other thing you should look at is "Gentility and the Middle Class." Look at the part where it talks about the middle class being able to buy toys for their kids (and bikes for themselves). This gave them a different experience than the lower classes had.