Dench, 71, was nominated Tuesday for an Oscar for her performance in the film, which was distributed by Weinstein's new production company, The Weinstein Co.
Weinstein, who co-founded Miramax Films, says Dench was turned down for interviews by NBC's "Today" show and ABC's "Good Morning America" and "The View."
"They said that she didn't fit their demographics," the outspoken movie mogul tells the magazine in this week's issue.
"I told that to my mother, who was pretty ... offended," he is quoted as saying. "I mean, what do they think, 25-year-old people can't watch 70-year-old people? The insanity of youth. It also assumes none of us like our families."
The International House of Publicity, which promoted "Mrs. Henderson Presents," declined to comment.
Bill Geddie, executive producer of "The View," told the AP that Dench had appeared a number of times on the show, including to promote "Shakespeare in Love." The actress won an Oscar for her performance in the 1998 film.
"I was never pitched Dame Judi Dench to appear on `The View' in support of `Mrs. Henderson Presents,' and we would welcome her back to the show anytime we have an opening," Geddie said.
Bridgette Maney, a spokeswoman for "Good Morning America," released a statement to The Associated Press that neither confirmed nor denied that Dench was turned down.
"`Good Morning America' welcomes a wide array of guests to the program each and every day. Judi Dench is a screen legend," the statement said.
A spokesman for the "Today" show didn't respond to a request for comment.