A New York State judge declared the state’s method of financing public schools illegal yesterday, saying it deprives New York City students of the sound basic education guaranteed by the state constitution. The judge handed the state legislature the delicate task of coming up with a new system by September 15 or facing judicial intervention. The judge found the school financing system also violated federal civil rights laws because it disproportionately hurt minority students. More than 70 percent of the state’s Asian, black and Hispanic students live in New York City. New York is the latest of more than two dozen states forced by courts to grapple with the job of assuring that poor students get an education that passes constitutional muster.