On weekends you can enjoy the playground, go-karts, a small train, a bouncy castle, and pony rides. These are available during different times throughout the day, but while we were there around noon we were able to take advantage of almost all of them. There is also a horse-drawn carriage that the whole family can ride. The best part is that all of these are completely free!
It's probably best to visit the racecourse with your kids when there are no races taking place, so as to avoid crowds and a lot of cigarette smoke. On these days, there is plenty of parking (also free), and lots of space on the grass around the track itself. It's a great place to have a picnic and watch the races in other parts of Japan being broadcast on the giant screen in the middle of the track.
Information and hours for the kids' activities: http://jra.jp/facilities/race/niigata/news/fureai.html
Location of the Niigata Racecourse: