This is "The Fox and the Forest", by Ray Bradbury, also released as "To the Future". From this review:
William and Susan Travis are vacationing in Mexico in October 1938, enjoying the local culture. Susan asks if they won't be found out. Bill tries to assure her, but she notices a man at a cafe they pass, a wide variety of liquor bottles arrayed on his table. Susan saw him earlier and believes he's from the Searchers, following them. William tries to keep her calm but Susan thinks of how it all started: her real name is Ann Kristen and her husband is Roger, from 2155 A.D., inhabitants of an Earth poisoned by war. When Ann's friend Rene mentioned a new travel service that allows people to go to different times for vacations, Ann saw the chance she and Roger had long hoped for, to escape their world and their jobs making bombs and disease weapons. The Kristens went to New York City in 1938 a month ago and on the third day fled to Mexico. Their first night in New York, however, they indulged themselves in all the luxurious drinks and smokes of the era, as the war economy of the Future made such things scarce. This kind of indulgence is what they noticed in the man watching them.
Melton invites them for drinks with his group in their hotel room. As they enjoy themselves, Melton tells the story of a man and woman who flee from 2155, immediately alerting the Travises to the real identity of this film company. William fires his gun, wounding one man; but it's too late and the camera is unveiled as the time machine that returns them to the future. The manager of the hotel was banging on the room door after hearing the shots but breaks in too late; the people seem to have disappeared into thin air. Later, while a priest blesses the room, the charwoman asks what's to be done with the contents of the closet, which holds a cornucopia of liquors and smokes.
I read this one as a child, in Alfred Hitchcock Presents Stories Not for the Nervous. After that, it was a matter of looking at the anthologies of Hitchcock that I'd read and looking at their story titles, doing a quick Google search on titles that seemed likely.
I found a copy online. The quote you remember:
A world that was like a great black ship pulling away from the shore of sanity and civilization, roaring its black horn in the night, taking two billion people with it, whether they wanted to go or not, to death, to fall over the edge of the earth and the sea into radioactive flame and madness.
And the detail Daniel Roseman recalled:
"Sorry." The stranger pulled up a chair. "Let us say I thought I knew you because you did not pull your trousers up. Everyone does. If they don't, the trousers bag quickly. I am a long way from home, Mr.-Travis, and in need of company. My name is Simms."