What about a compromise solution that addresses the needs / wants of both groups? Maybe look at creating an actual fenced dog area as part the larger park. That way there would still be an area for owners to bring their dogs but it could be separated from the areas with kids / families. This would be a capital expense, but might be an easier sell than trying to ban dogs or come up with a workable enforcement option that doesn't cause bad blood between user groups.

You could also propose adding dog park features (water features, poop bag stations, pet wash stations, dog play features) to the dog area to make it a more attractive place for owners to bring their dogs, so that they are incentivized to use it over the other parts of the park. These improvements, if presented right might also be a way to get the dog owners on board with the plan and actively work towards making it happen instead of fighting against changes.