All dogs must have a valid license within their local (home) municipality. For most members, this means that your dog must be licensed in the city of Beacon since that is where most members reside.
Membership is also open to people who live outside of Beacon (Cold Spring, Glenham, Wappingers Falls, Fishkill, etc.); in such cases, your dog must be licensed in those respective municipalities - no exceptions.
The Beacon Dog Park is not a city run dog park, it is built and operated by a volunteer organization (501/c3) called Friends of Beacon Dog Park Inc, as such it can can not provide any assistance with licensing your dog - it must be done at a town or city hall.
Here are some helpful links about getting a valid license in your particular town: