If you do go to a Yankees game at Yankee Stadium, then do not take any bookbags or large purses or any other large containers with you. They are not allowed in under any circumstances, and they will not let you in unless you get rid of them.
You can store them at a bowling alley on River Avenue, but they charge about $5 for this simple service, and it takes a while after the game to get your stuff back due to lines.
The Bronx is just like any city with the same social problems (drugs, theft, petty crime, homelessness etc.). I consider it a 'watch you A--' area, similar to visiting Newark, NJ, Philly, or Harlem. Don't skip it soley because of a fear of crime. If you want to visit, do so. Just don't act like a fool.