I was thinking about which major sports could allow men and women to participate together. Most sports, such as NFL football or NBA basketball, would be completely dominated by men. In the case of NFL football, you could maybe make the kickers be women, but that's about it, and it would reduce the entertainment value of the game since the kickers wouldn't be able to kick as far.
But baseball is a different story because the players have drastically different roles (i.e. pitcher, catcher, etc.). You could have the infielders be women (1st base, 2nd base, 3rd base, shortstop) since they don't have to throw as far, with the remaining positions (right field, center field, left field, catcher) being men. For the pitcher, there could be 2 pitchers on the field (one woman and one man), where the woman pitches to women and the man pitches to men. The pitcher which is currently not pitching would be the only one allowed to field the ball. This makes the number of male players the same as the number of female players.
Also, baseball teams already have a balance of home run hitters and reliable (high on base percentage) but less powerful hitters. Since men tend to be stronger than women, most of the home run hitters would be men and most of the reliable hitters would therefore be women, although some teams might mix this up a little bit if they have an exceptionally powerful female hitter.
Despite the characteristics mentioned above, the game still favors men more than women, because of home runs, etc. To remedy this, cheerleading will become part of the game. If you go to a baseball game, between innings there is already a lot of music pumping and T-shirt guns firing and audience giveaways and stuff to keep people entertained. Instead of this, after each half-inning, there would be cheerleaders for one of the teams performing a highly acrobatic cheerleading routine the duration of a song, and after the inning (i.e. after both teams perform a cheer), judges would award the best routine a point.
Since with the cheerleading the innings will be longer, there could be 5 innings instead of 9 innings. The guaranteed 5 points from the cheerleading will be added to the points from baseball to generate a total score, and the highest total score would win.
An interesting (but not required) twist on this would be that there would be a maximum number of overall players on a team (including both baseball and cheerleading) that is low enough to require some of the players have to both play baseball and do cheerleading. I can picture a scenario where one of the cheerleaders gets injured and one of the manly home-run hitters has to take their place to avoid forfeiting the game.
A final idea would be that the teams do not have a home city. The teams would still have a name and an identity that they use to try to attract fans, but they play in a different city each time (for the most part). The upside of this is that you wouldn't have to build them stadiums, and the size of the stadiums they play could fluctuate (minor league vs. major league vs. maybe even large high school stadiums) depending on the popularity of the team and the sport, thus drastically reducing the financial risk.