At a Coney Island park this summer, children in a basketball summer camp wore blue-and-white uniforms bearing the logo of “The Coney,” a proposed casino project in the neighborhood and the camp’s sponsor.

A few months earlier on Long Island, where Las Vegas Sands is pitching a casino, children from local soccer teams were invited to a Sands-sponsored training session featuring the superstars David Beckham and Carli Lloyd.

In the escalating race to be one of the first to open a casino in the New York City area, developers are rushing to win local support after state regulators said projects needed to be “embraced by the community.” A casino logo emblazoned across youth sports jerseys is just one example of the lengths to which gambling operators have gone to woo those communities and secure a coveted license.