Parks become safe when the community USE them- including dog walkers.
My local park used to be a bit of a no-go area, but it has had a lot of money spent on it in the ten years I've lived here, and there's a community group which runs events and fund-raises. So in this not-especially salubrious area, we have evening Bat Walks, Picnic Saturdays, occasional "waggiest tail"-type Dog Shows, with Police Dog Display Teams etc, Easter Egg Hunts, and lots of other stuff throughout the year.
The play areas are fantastic, all nature-based adventure equipment and soft turf, something for all age groups.
We still have the bowling club, and the lovely Victorian landscaping which has been preserved. Trees that have had to be cut down are decorated by the most amazing hand carving from a local artist.
There are outdoor exercise machines dotted around the park randomly, and you see people of all ages using them.
There's a safe fenced-in field for people to exercise unsociable or escape artist dogs, but the rest of the park is still available to them, and the children's play areas are only visually demarcated by plants rather that fenced-in, yet people keep their dogs away.
I won't pretend that I've never seen dog poo anywhere there, but there are lots of bins, and the vast majority of people seem to pick up their dogs' waste.
This isn't in leafy Surrey, it's not an affluent area, but because the planning and efforts to regenerate it has taken into account all the groups of people who would like to use the park, including dog walkers, everybody seems to rub along very happily.
If a park excluded dogs and their walkers, I agree with the OP that we would lose a core of community-spirited users who are there day in, day out, and contribute to a safe environment for the children who come out on sunny days. The key, I think, is cooperation and inclusion, not banning dogs.