Looking over my small collection of Titanic books, Walter Lord, the dean of Titanic scholars, provides the fullest answer to your question through his assembly of eyewitness accounts. Small clues alerted the Carpathia’s passengers that something was amiss in the early morning of April 15. One passenger smelled coffee brewing around 1:00 AM; another noticed that, contrary to custom, all the ship’s lights were burning. Other people heard the sounds of lifeboats being unhooked, which made some of them conclude that the Carpathia was in trouble; others noticed that the ship’s engines had revved up, rattling the furniture in staterooms. Cold water came out of the hot water taps in the rooms; the ship has diverted all its steam to feeding the engines. But the bitter cold was perhaps the main clue that something was wrong. Carpathia and its 725 passengers were headed for the Italian coast; it should have been getting warmer at sea. Louis Ogden cornered a quartermaster and asked what was wrong. “We’re going north like hell,” he answered. “Get back in your room.”

When the ship reachedTitanic’s last reported location at 4:00 AM, its passengers started to assemble on deck and watch quietly as the wreck’s survivors pulled aside in lifeboats; they were puzzled about what they were seeing. One woman thought the lifeboats were from a downed dirigible. Lord writes that “Strangest of all was the silence. Hardly a word was spoken. Survivor Lawrence Beesely concluded “they were simply in the presence of something too big to grasp.”

Carpathia’s passengers didn’t have any duties per se, though some had their assigned rooms rearranged to accommodate Titanic’s first-class passengers and linens destined for second class got redirected to Titanic’s first-class folks. (Two of these first-class passengers willingly offered some of these to a second-class survivor Juliette LaRoche to make diapers for her two babies—she had lost her husband.) Even in a disaster class prevailed. But passengers helped as they could. Lord again:

”The Carpathia’s passengers pitched in gallantly—digging out extra toothbrushes, lending clothes, sewing smocks for the children out of steamer blankets brought along in the lifeboats. A Macy’s wine buyer bound for Portugal became a sort of guardian angels for the three Gimbel’s buyers. Mrs. Louis Ogden took cups of coffee to two women in gay coats and scarfs sitting alone in a corner. “Go away,” they said, “we have just seen our husbands drown.”

Sources: Walter Lord, A Night to Remember (If you don’t know this short classic, you should read it. It’s from 1955 but in many respects is still the best, most vivid retelling of the event.) Charles Pellegrino (one of the living Titanic experts), Farewell Titanic, Her Final Legacy (2012), John Welshman, Titanic, The Last Night of a Small Town” (2012).

If you can’t tell, I’m a bit of a Titanic nut so if you haven’t read it, let me stress that Walter Lord’s A Night to Remember really is the classic account. He interviewed about 60 survivors and drew on their written accounts to write it. It’s short—only 160 pages of text—and vividly told, an absolute page-turner. He gets the mechanics of the sinking wrong, mischaracterizing the damage the iceberg did and rejecting the idea that the ship broke in half (which it did) since he was writing 30 years before Ballard discovered the wreck in 1985; but his storytelling is unsurpassed even by more recent books. I should add that the 1958 film adaptation of the book, also called A Night to Remember, is a much better, more accurate, and dramatically satisfying recounting of the Titanic story than James Cameron’s 1997 version.