Or alternatively, keep the kids on harnesses to stop them picking up poop and banish them to the Triangle where there's already a perfectly good swing park to stop the neurotic non-childrearing majority from being disturbed by squealing kids.
The majority of both parents and dog-owners are polite, pleasant, clean up after their respective charges and don't cause a problem to other users of the park. It seems harsh to penalise either of these groups and banish them to the less pleasant surroundings of the triangle for the sins of a few.
Perhaps we just need an effective way to reporting anti-social behaviour so they can be removed from the park. Ye old park-keeper perhaps?
I do use the Horniman Gardens regularly in the summer and can honestly say that I wasn't caused stress by any dog-owner. Any poop was promptly picked up and they never rampaged through my picnic. On the other hand, I did have a few accidentally mishit footballs whack me but, tongue-in-cheek first paragraph aside, I wouldn't suggest banning kids as a result. The Gardens are public places and I don't expect the same level of privacy and behaviour as I'd get in a private garden.