Rashana Jackman isn’t old enough to vote in an election, but she could soon have a vote on a city-appointed board that takes influential stands on neighborhood issues. At 17, the Brooklyn high school junior is considering applying to serve on her community board, under a new state law that lets 16- and 17-year-olds join the panels that function as front lines of local government in the nation’s biggest city. The advisory but oft-heeded groups opine on zoning changes and liquor license applications, consult on city budgeting for local projects and serve as conduits for community concerns. “It’s a great...