Every year the Heart of America Shakespeare Festival is held in Southmoreland Park (literally acroos the street from the Nelson-Atkins Art Museum) in June and July. The play is free. You can bring chairs or blankets and picnics to the play. They do have some food and drink vendors and portable bathroom facilities. We always bring blankets, pillows, drinks and fun snacks. The plays start around 8:00pm so this even is for older kids or younger kids who can take the late hour/love the Bard and wouldn't miss it. It is literally in the park, so it can be called off due to weather, but it's rare.
The park is clean, well lit, and safe and there is lots of off street parking on all sides of the park. Get there a little early for the pre-play show. There is a small playground for younger kids, but you are unable to see both the play and play ground area at the same time.