The author of the story acknowledged the criticism and said it was just an “oversight.”
The New York Times has posted several stories about the Menendez trial, including one by the same author a week or so ago that mention the fact that he's a Democrat. It seems quite likely that it was just an omission as the author said, not part of some conspiracy to hide the fact that he's a Democrat.
The Times reported on the FBI investigation of Menendez as far back as 2013 before he was even indicted. That article mentioned his party affiliation as well.