But as trailgirl pointed out, children can be there with dogs and have some familiarity with the species. They might need some help if a dog freaks out when it sees them but then I think even in a dog park you do have that responsibility if your dog has issues with kids/men/wheelchairs. The child should have a parent to help and supervise.
I am not advocating for kids to take over but OTOH I think that kids are usually part of dog ownership in a family so a complete ban doesn't actually make a lot of sense either. And there should be balance to the dog supervision as well. If your dog has issues with any segment of the population, even kids, then even in a dog park you are going to have to be extra careful and watchful.
Dogs who are prone to dangerous freak outs should not be in a dog park. Most parks I know of do have rules to that effect.
I've seen well supervised
kids around 8 and above who can, indeed, handle any reasonable situation with dogs. But the OP, and several posters, discussed parents who did not supervise their kids ... or who brought kids but no dog ... or even worse, just dropped off kids and dogs and went their merry way!
It is also true that supervising an active toddler in a park full of random, unleashed dogs may be well nigh impossible. Many active toddlers will naturally tend to act like prey -- running and screeching.
Toddlers and strollers are especially vulnerable to over-friendly dogs of any size. They just get bowled over!
Even the best-behaved, unleashed dogs do not belong in a tot lot. In fact, many tot-lots also -- wisely -- prohibit leashed dogs. Similarly, dog owners in an area set aside for unleashed dogs should not have to supervise a random child as well, nor should they be forced to leash their dogs because there are small children present.
One toddler running around can, in effect, take over a park.
While it can be difficult to arrange, the best policy is for adults to team up, and take turns with the kids on the other side of the enclosure.