"And we are trying to do what's right even when Donald Trump won't".
Moving to squelch internet speculation, Pence's spokesman said the vice president did not write the article.
"They don't like Donald Trump and I don't like them because they're very dishonest people", the president told reporters.
She said the press should be fair, unabashed and responsible.
Her comments came as Trump's top lieutenants scrambled on Thursday to deny authorship of the explosive op-ed that has plunged his presidency into its worst crisis yet by proclaiming a secret insider resistance to his "reckless" and "amoral" leadership.
So far, Trump has publicly supported Mattis, telling reporters on Wednesday the defense secretary will stay in his current job and he believes the quotes attributed to Mattis in Woodward's book are fabricated. In one tweet, he demanded that the newspaper name the source for legal action.
"The dilemma - which [the president] does not fully grasp - is that numerous senior officials in his own administration are working diligently from within to frustrate parts of his agenda and his worst inclinations", wrote the official.
The New York Times says the article was written by a senior official in the Trump administration. In this case, former White House chief economic adviser Gary Cohn swiped the draft letter off the Oval Office desk to prevent Trump from signing it, terminating a critical trade agreement with South Korea.
Trump described the writer of the opinion piece as someone "probably who is failing and probably here for all the wrong reasons". That measure defines methods to legally remove a president from office.
The president called the individual "GUTLESS" on social media, and White House spokesperson Sarah Sanders called the anonymous writer a "coward".
That would be a problem for the Times, partly through no fault of its own, said Kathleen Hall Jamieson, communications professor and director of the Annenberg Public Policy Center at the University of Pennsylvania.
Speaking to TMZ in NY on Thursday, Kelly said the anonymous White House official should "grow up" and "take a stand".
It concluded: "There is a quiet resistance within the administration of people choosing to put country first".
The op-ed also continued a pattern of leaks and betrayals from within the administration that have contributed to several unflattering exposes, including Michael Wolfe's "Fire and Fury" and Bob Woodward's "Fear" that describes the Trump White House as "crazy town". Members of his cabinet rushed to deny authorship in an unedifying spectacle reflecting the depth of suspicion gripping the Trump administration.