Like the Supreme Court, the Fed has considerable independence from day-to-day politics. It has seven governors who are appointed by the president and approved by the Senate. They can serve 14 year terms, although most do not stay for the full period.
The Open Market Committee that sets interest rate policy also includes the twelve district bank presidents. Five of these twelve bank presidents have a vote at any point in time, although all twelve take part in the discussion. The bank presidents are appointed through a somewhat opaque process that has historically been dominated by the financial industry, although this process was opened up somewhat during Janet Yellen's tenure as Fed chair.
This process insulates the Fed from the whims of the president and other political figures. However, there is nothing inappropriate about the president or any other elected official commenting on Fed policy, as The New York Times implies in this piece.
Given the close ties of the bank presidents, and often the governors, to the financial industry, a Fed that is considered off limits for political debate is likely to be overly responsive to the concerns of the financial industry. This is likely to mean, for example, excessive concern over inflation and inadequate attention to the full employment part of the Fed's mandate. It is understandable that the financial industry would like to keep the public unaware of the importance of the Fed's actions, but the public as a whole does not share this interest.