Perhaps the first answer I would give to this question would be "Not as bad as video games are for your son." At least most movies have story lines and characters that can be compared and discussed in relation to real life situations, rather than depicting enemies as mere targets. But the movies you and your children watch are filled with sexists messages that can have effects on young girls.
With all the attention given to film content because of the "#Metoo" and "TimesUp" movements, I've been pondering what I was exposed to as a child. The sad conclusion I came to was that although what I was exposed to, in the 1970s, was bad, today's films aren't much better and possibly are worse.
I wasn't bothered by the "Bond girls," back when I was 12, because they were mostly heroines and even action heroines in some circumstances: Think of "Pussy Galore," an air commander with a fleet of all-female pilots--almost a radical feminist statement! (if they hadn't been mostly 22- year old blondes). Or Tully Masterson, who tried to avenge her sister's death with a sniper rifle. Most of the sex was depicted by suggestion and innuendo. Today's action movies are either cartoon/comic adaptations that have no relationship to real life, or, they are live action movies with male heroes (Liam Neeson, Vin Diesel, Tom Cruise, Daniel Craig) in which the female roles aren't memorable. The last Bond movie I saw, "Skyfall," was execrable and featured females as failures and quick victims, but was dressed up with Judi Dench in the minor role of "M." And because Daniel Craig was tied up filming that movie and the next Bond film (which needless to say I did not see), the sequels to "Girl with a Dragon Tattoo"--an immensely popular story with an unusual heroine and a feminist message--did not get made.
An insidious message that has persisted and perhaps even gotten worse is the message that women can't be friends with each other. I've noticed this even in recent, so-called feminist films: Whereas in the "Hunger Games" books Katniss Everdeen had a female friend, this small role was excised from the films. The cinematic Katniss has no female friends. In "Maleficent," purportedly a feminist twist on the "Sleeping Beauty" fairy tale, Aurora plays with seemingly male forest creatures, Maleficent's companion is a male crow, and the three fairies who raise Aurora are depicted not as friends but as a group of catty, quarreling women who insult each other. As in the Hunger Games movies, the positive female relationships are family relationships--mother/daughter, stepmother/daughter, sister/sister. The message is that outside of the family, other women can't be trusted.
Parents might want to think about another message besides misogyny that their daughters might grasp from misogynistic films: The message that adults aren't role models. I'll never forget my response to the movie "M*A*S*H" when I was around 11 or 12: When the "heroes" humiliate the "Hot Lips" Houlihan character by ripping her shower curtain off and exposing her to a crowd, I realized this action was supposed to be cheered by the audience. Since I knew what they were doing was wrong, the message I received was that adults didn't care about the difference between right and wrong and were a bunch of hypocrites. These type of movies, which are still extremely common, should not be viewed by children under 15. In fact, they should not be viewed at all.
What children need is to see are depictions of other children with whom they can identify, who have friends (including same-sex friends) and who solve problems, and depictions of adults who are icons or role models. Today's movies and TV don't cut it. You can solve this problem by encouraging your children to read books.