"The Cage", a short story by A. Bertram Chandler, originally published in The Magazine of Fantasy and Science Fiction, June 1957, available at the Internet Archive, and in Authentic Science Fiction #82, July 1957, also available at the Internet Archive. Reprinted many times; does any of these covers ring a bell? The anthology you read it in in the early/mid 1970s might have been 12 Great Classics of Science Fiction (edited by Groff Conklin), or The Penguin Science Fiction Omnibus (edited by Brian Aldiss), or *Anthropology Through Science Fiction (edited by Carol Mason, Martin Harry Greenberg, and Patricia Warrick).
Atomic engined passenger starliner suffers major engineering problem, hurriedly sets all passengers and a few crew down on nearest world. Starliner then explodes in space, marooning the passengers and a few crew
Right, except that the starship exploded on the ground:
At least two hundred days had passed since their landing on the planet without a name—an unintentional landing made when Lode Star's Ehrenhaft generators, driven far in excess of their normal capacity by a breakdown of the electronic regulator, had flung her far from the regular shipping lanes to an unexplored region of space. Lode Star had landed safely enough; but but shortly thereafter (troubles never come singly) her pile had gone out of control and her captain had ordered his first mate to evacuate the passengers and those crew members not needed to cope with the emergency, and to get them as far from the ship as possible.
Hawkins and his charges were well clear when there was a flare of released energy, a not very violent explosion. The survivors wanted to turn to watch, but Hawkins drove them on with curses and, at times, blows. Luckily they were upwind from the ship and so escaped the fallout.
on unexplored, rainy jungle world.
The climate was against them, for a start. Hot it was, always in the neighborhood of 85o Fahrenheit. And it was wet—a thin, warm drizzle falling all the time.
Wet environment pretty much is survivable, but wet rot and fungus effectively destroy all clothing.
The air seemed to abound with the spores of fungi—luckily these did not attack living skin but throve on dead organic matter, on clothing. They throve to an only slightly lesser degree on metals and on the synthetic fabrics that many of the castaways wore.
Looking at the long haul, one of the female passengers has two interested suitors, one a fellow male passenger, one is(maybe) starliner 3rd officer. Ship's doctor (classic elder type) referees boxing-type winner-gets-the-girl match.
"All right, then," said the girl. "Monogamy; I rather prefer it that way myself. But I warn you that if that's the way we play it there's going to be trouble. And in any murder involving passion and jealousy the woman is as likely to be a victim as either of the men—and I don't want that.
"What do you propose, then, Miss Hart?" asked Boyle.
"Just this, Doc. When it comes to our mating we leave love out of it. If two men want to marry the same woman, let them fight it out. The best man gets the girl—and keeps her."
"Natural selection . . ." murmured the surgeon. "I'm in favor—but we must put it to the vote."
[. . . .]
Hawkins nodded. He looked at the four in the center of the arena—at the strutting, barbaric woman, at the pompous old man, at the two dark-bearded young men with their glistening white bodies. He knew them both—Fenner had been a Senior Cadet of the ill-fated Lode Star; Clemens, at least seven years Fenner's senior, was a passenger, had been a prospector on the frontier worlds.
[. . . .]
"No gouging, no biting!" called the doctor. "And may the best man win!"
Alien starship shows up overhead, scoops up Doc and both suitors and the girl.
Hovering above the arena was a helicopter. There was something about the design of it, a subtle oddness, that told Hawkins that this was no earthly machine. From its smooth, shining belly dropped a net, seemingly of dull metal. It enveloped the struggling figures on the ground, trapped the doctor and Mary Hart.
Along with a patch of their jungle, and re-creates their 'native' environment aboard the alien starship.
The world to which they were taken would have been a marked improvement on the world they had left, had it not been for the mistaken kindness of their captors. The cage in which the three men were housed duplicated, with remarkable fidelity, the climatic condition of the planet upon which Lode Star had been lost. It was glassed in, and from sprinklers in its roof fell a steady drizzle of warm water. A couple of dispirited tree ferns provided little shelter from the depressing precipitation. Twice a day a hatch at the back of the cage, which was made of a sort of concrete, opened, and slabs of fungus similar to that on which they had been subsisting were thrown in. There was a hole in the floor of the cage, which the prisoners rightly assumed was for sanitary purposes.
Despite their best efforts to communicate, Pythagorean theory twig diagrams and more, aliens keep treating them as zoo-specimens.
". . . Try Pythagoras' Theorem again," he said to the cadet.
Without enthusiasm the youth broke fronds from the nearest tree fern. He broke them into smaller pieces, then on the mossy floor laid them out in the design of a right-angled triangle with squares constructed on all three sides. The natives—a large one, one slightly smaller, and a little one—regarded him incuriously with their flat, dull eyes. The large one put the tip of a tentacle into a pocket—the things wore clothing—and pulled out a brightly colored packet, handing it to the little one. The little one tore off the wrapping and started stuffing pieces of some bright blue confection into the slot on its upper side that, obviously, was a mouth.
"I wish they were allowed to feed the animals," sighed Hawkins. "I'm sick of that damned fungus."
One 'night' sleep period, rainforest alien rat wakes girl up running across her legs and she demands that the men Do Something about it.
"I . . . I don't know . . . Something small, sith sharp claws . . . it ran over me . . ."
[. . . .]
"He must be the local equivalent of a mouse," said the doctor, "although he looks nothing like one. He comes up through the floor somewhere to look for scraps of food. We're trying to tame him—"
"You encourage the brute?" she screamed. "I demand that you do something about him—at once! Poison him, or trap him. Now!"
They manage to fashion a trap, and catch the rainforest alien rat. They decide to keep it as a pet and feed it some of the fungus.
But Joe was not killed. The three men were rather attached to him. With the coming of daylight they transferred him to a cage that Hawkins had fashioned. even the girl relented when she saw the harmless ball of multicolored fur bouncing indignantly up and down in its prison. She insisted on feeding the little animal, exclaiming gleefully when the thin tentacles reached out and took the fragment of fungus from her fingers.
At this point, the aliens running the starship figure out that the humans aren't primitives, oops, and set course for the nearest human Space Patrol base.
"Our hosts have apologized very sincerely, and they have more suitable accommodations prepared for us. Then, as soon as they have a ship ready, we're to go to pick up the other survivors."
When asked what changed their minds, they are chagrined to admit that it is well known that only civilized beings keep pets.
"Not so fast," said Boyle. "Put us in the picture, will you? What made them realize that we were rational beings?"
Hawkins' face darkened.
"Only rational beings," he said, "put other beings in cages."