It is perfectly legal to do background checks on potential employees.
Criminal Court records are public information in New York. In fact, New York State provides This Site to look up anyone sentenced to state prison since the early 1970's. Check the site for details on who is in the database.
With respect to credit checks, you should read the details of the Fair Credit Reporting Act on This Site Part 3 &5.
Also be aware that under New York State's Human Rights Law (Executive Law § 296[a]) it is unlawful to refuse to hire someone based on age, race, creed, color, national origin, sexual orientation, military status, sex, disability, predisposing genetic characteristics, or marital status.
I hope this information helps.
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