It really depends on the genre, they’re so different.
For modern espionage games, it’s easily the more serious James Bond movies, The Sandbaggers, and action films of the 1980s. It’s what I was watching while cutting my teeth as a GM. It taught me the three act style of storytelling, but also gave me the “action movie” style of plotting an adventure — pick three exotic locales, put together an action set piece unique to them, string the story through. These differing styles — the over-the-top Moore era Bonds and the more realistic Sandbaggers and mystery/spy novels gave my espionage games a high-level of verisimilitude, but we still had big chases, fights, and the product-placement Bond lifestyle.
For science fiction, it was Blade Runner and Road Warrior, more than anything else until Babylon 5 taught me how to do story arcs well. There’s also been a heavy transhuman influence since I read Kurzweil’s The Age of Spiritual Machines in the 1990s. In the aughties, the more “realistic” sci-fi of Battlestar Galactica dovetailed well with my attempts to make Star Trek less clumsy and utopian (read, boring.) It was about that time I went back and watched the best of the Trek series, Deep Space 9. I also learned that big battles are often less important than the human impact (although the simulation-heavy gamers would disagree.) A good example of this would be The Man From U.N.C.L.E. currently on screen. There’s a big chase piece that we see in reflections on a windscreen as the lead character relaxes and reflects on his situation. It’s funny, but it also is subversive for an action movie. They do it again with the big secret base invasion, which is told in a series of split screen cuts that give you what you need, but don’t bog down the story.
In Victorian science fiction, funnily, none of the speculative fiction of the period! Mostly, my inspiration came from the actual history of the time and westerns. Every time I do historical games, i find I tend to weave the characters into real events and let them play. My main inspiration for that style of storytelling was the Flashman series of novels by the late George MacDonald Fraser.
Pulp games set in the ’30s have a lot to thank from Raiders of the Lost Ark and similar movies, especially once I started to mash them up with James Bond and other “pulp” movies and books.
Supers games owe a lot to Watchmen, The Dark Knight Returns, and some of the angsty X-Men stuff of John Byrne and early Chris Claremont, but now also the influence of Marvel cinematic universe.