People are working hard to diversify New York's elite public high schools, such as Stuyvesant, but their efforts are falling short. According to The New York Times:

Among the 21,490 public school students who last year took the exam, the single gateway to eight high schools, 6 percent of blacks and 7 percent of Hispanics were offered admission, compared with 35 percent of Asians and 31 percent of white students. The disparities were the worst at Stuyvesant, where 2 percent of blacks, 3 percent of Hispanics, 24 percent of whites and 72 percent of Asians were accepted.

My guess is that this is more of a class issue than a race issue. Many Black and Hispanic children come from low-income families and can't really afford tutors and books. Additionally, a lot of their parents have a job, sometimes two, and can't always be around to supervise and make sure their children are studying hard. I think it's the same reason as to why Ivy League universities are predominantly White.

Also, a lot of these kids lack the motivation. They see their parents working two jobs, struggling to make ends meet, and they think, "I'm going to end up like that." Sometimes it works the opposite, where kids will see the hard lives their parents lead and will vow to make a better life for themselves, but that isn't always the case. It's a class issue. Upper-middle class White and Asian children have the privileges that support their acceptance into these elite schools; Hispanic and Black children generally don't. It's a vicious cycle.