Translation:Do you want to go to New York?
No; the given exercise translation is correct: 你们想不想去纽约? means "Do you want to go to New York?" - a positive question, not a negative question. If you want to say "Don't you want to go to New York?" in Chinese, you could use the translation that Imnuts7 provides: 你不想去纽约吗?
To ask an affirmative/negative "yes/no" question in Mandarin, you can use the "verb ... 吗“ form or the "verb 不 verb" form. In the case of compound verbs (verbs that require more than one character), often just the first character is "negated," e.g., to say "Do you know?" you can say 你知道吗? or you can say 你知不知道? You do not need to say 你知道不知道?