This particular park is a bit tricky. In the attempt to sort this out, I put in a Freedom of Information request to the Town of New Castle about a year ago. For all intents and purposes, Battery Park seems to operate as a public park. The Town of New Castle Police Department patrols the area and issue tickets for violations. However, the park lands seem to be owned and managed by The Trustees of the New Castle Common which is a non profit organization first incorporated in 1764 and reincorporated by the Delaware Assembly in 1792. The Trustees of the New Castle Common is a completely separate entity from the Town of New Castle.
I'm not a lawyer, so I can't decode exactly how this all pans out for public versus private lands. The prohibition against firearms isn't an official ordinance by the Town of New Castle, but, rather, a rule voted into place by a private, non governmental, non profit agency, board of trustees. The best I can determine is that because it isn't an ordinance or law, it seems to be just another "no guns sign" without force of law.
However, because the property seems to considered private property, open to the public, one could be asked to leave if they were discovered to be carrying, or one could be charged with criminal trespass. I got exactly ZERO cooperation from the either the New Castle County Police Department or the New Castle County Town Hall. Each referred me to the other which resulted in me filing the Freedom of Information Request. To the best I can determine, there doesn't appear to be any force of law behind the "rule" posted at the park. Once again, I'm not a lawyer...