To get your free access, go to the BHPL web site and click All Things E, then click the link to the New York Times. You will need your library card's barcode number, and you will also need to have a login for the New York Times web site. I had one from back when the New York Times was free as long as you logged in, but if you don't have a login, just click Register.
The New York Times Book Review is kept on a stick. Fun newspaper stick fact: you are allowed to take the publication off the stick while you read it. Fun newspaper stick fact 2: our library regulars are evenly divided between those who leave the stick on and those who take the newspaper off. Which side are you on?