“It just doesn’t make sense to delay thousands of people over one sick passenger,” she said in an interview. “There has to be a better way to handle it.”

The response by the authorities depends on the condition of the sick passenger. Workers are instructed not to move riders who cannot walk off the train on their own and to wait for emergency medical workers to assess the patient. If the passenger can step off the train, someone else — a friend, a worker or a police officer — must stay until help arrives. Otherwise, the conductor must stay with the rider, and the train may have to be pulled out of service.

In 2013, Casey Blue James was standing on a crowded train in Queens when she started to feel woozy and fainted. When she awoke, the train was stopped and everyone was staring at her. A transportation authority worker stayed with her at the station until an ambulance arrived to take her to the emergency room.

When Ms. James, 26, who works in publishing and lives in Jackson Heights, Queens, had the same dizzy feeling last year, she shoved her way off the train to avoid embarrassment and fainted on a platform. After being told she had a condition in which her blood pressure can drop rapidly, she now carries water and snacks in her bag.

Before it happened to her, she was annoyed by the delays like everyone else.

“I feel much more sympathetic when I hear that’s the reason now,” she said.

In an ad campaign, the authority has urged riders who feel sick to get off at the next stop to get help — and to alleviate delays. Officials have reminded riders not to pull the emergency brake, which abruptly stops the train, because that can slow the emergency response if the train is stranded between stations.

Starting in the late 1990s, the transportation authority tried to reduce the impact of sick passengers by stationing nurses in a few busy stations. But the program ended around 2008, done in by a budget crunch.