You should begin your search in ProQuest Newsstand. Limit your search by publication and choose New York Times Book Review. This database provides coverage from Jan 5, 1986 - present for the NYT Book Review.
We keep back issues of the New York Times Book Review for 6 months in the periodicals area. Ask the Circulation Desk staff for assistance if you want to browse through recent issues.
You might also search the Lexis Nexis database. Choose "Search the News". Then, in the 'by Source Title" box, limit your search to the New York Times. Use the term "book review" in your search string.