Before the bell Thursday, The New York Times reported third-quarter earnings that beat on the top and bottom lines. The company also reported more than 3 million digital-only subscriptions and 4 million total subscriptions as of the end of the quarter.
The stock jumped on the news, hitting a new 52-week high. It was up more than 6 percent in afternoon trading.
Trump, who has recently stepped up his attacks on the media, has used Twitter to repeatedly go after The New York Times.
Thompson called the president's tweets "free publicity" and a "marketing boost."
However, while the paper saw about 400,000 more subscribers than expected in late 2016 and early 2017 because of "intense interest and anxiety about the election," that petered out by spring 2017, he said.
Now it is seeing what he believes is "business as usual."
Thompson also acknowledged that we are living in incredibly polarized and controversial times.
A new Politico/Morning Consult poll conducted over the past few weeks shows 62 percent of voters think the national news media has done more to divide the country than unite it. Of those surveyed, 56 percent said Trump has done more to create division than unity.
Earlier in the week, Trump blamed the "fake news" for the nation's divide.
"With so much mud flying around ... of course I understand why people are looking for someone to blame," Thompson said. "We're just trying to do our jobs as journalists and editors and marketers, and all the rest of it."
However, he said public support for high-quality journalism is strong and growing.
"We live in a world which is very frightening and very hard to understand," Thompson said. "All around the world there are readers who want to understand what's happening and they want to support independent, without fear or favor, journalism."
— CNBC's Sara Salinas contributed to this report.