Children are welcome at all major racetracks, Paula, as long as they are accompanied by a parent or adult guardian. As Sandra said, virtually all tracks are family oriented, and many of them actually have special days that are oriented and designed around young spectators and fans. Children under 18 cannot place wagers, however. And at those tracks which have on site casinos ( which includes Oaklawn) minors under 21 years of age are prohibited from entering the casino floors. But kids are welcome at all other areas of the track, as long as there are adult guardians or parents present.
Didn't you watch any of the Triple Crown races last year? If you had, you would have seen that American Pharoah's trainer, Bob Baffert, was never alone at any of the races. He was always accompanied by his wife, Jill, and their 10 year old son, Bodie. The same thing happened back in 2006, when Barbaro was running. HIS trainer, Michael Matz, was always accompanied by his son Alex, who at that time was 9 years old. ( Alex is now a college aged young adult.)
I personally have made multiple trips to Delaware Park, which is close to where I live- and every time I've been there, there are lots of kids around in the paddock and in the grandstand. No, the kids aren't on the casino floor, because they're not allowed in there- but the restaurants and other facilities at the track welcome children. You should have no problem with getting your son or grandson into Oaklawn, Paula.