I'm looking for the title/author of a short story in which humans are living on a ship on another planet and they let in a human-looking male alien. The alien is (possibly) red-headed and very good-looking. The men on the ship all get sick, go through some sort of metamorphosis, and end up looking like the alien. They can then live outside the ship. Then a female alien comes up to the ship and the women on the ship have to decide whether to let her on or not.
I'm looking for the title/author of a short story
"Contagion", a 1950 novelette by Katherine Maclean, also the answer to the question Short story: planet with wrong chirality and an unaccepted answer to the question Chirality and Colonists. You can read it at Project Gutenberg or the Internet Archive, or listen to a reading at LibriVox or YouTube, or you can read a review by Doomsdayer.
in which humans are living on a ship on another planet
The people of the ship waited while their doctors, in airtight spacesuits, hunted animals to test them for contagion.
and they let in a human-looking male alien. The alien is (possibly) red-headed and very good-looking.
Not an alien, he's a descendant of human colonists:
This one looked like a man, a magnificently muscled, leanly graceful, humanlike animal. Even in its calloused bare feet it was a head taller than any of them. Red-haired, hawk-faced, and darkly tanned, it stood breathing heavily, looking at them without expression. At its side hung a sheath knife, and a crossbow was slung across one wide shoulder.
[. . . .]
"Patrick Mead is the name," smiled the man, shaking hands casually.
The men on the ship all get sick,
June rose. It was time for her to take over the job. She leaned across his desk and spoke into the speaker system. "Doctor Walton talking," she said. "This is to the women. Don't let any of the men work any more; they'll kill themselves. See that they all go into the tanks right away. Set the tank dials for deep regeneration. You can see how from the ones that are set."
go through some sort of metamorphosis, and end up looking like the alien.
He turned his head and looked at his arms and legs, pale blond arms and legs with a down of red hair. He touched the thick left arm, squeezed a pinch of hard flesh. "It isn't mine," he said, surprised. "But I can feel it."
Watching his face was like watching a stranger mimicking and distorting Max's expressions. Max in fear, Max trying to understand what had happened to him, looking around at the other men sitting up in their tanks. Max feeling the terror that was in herself and all the men as they stared at themselves and their friends and saw what they had become.
"We're all Pat Mead," he said harshly. "All the Meads are Pat Mead. That's why he was surprised to see people who didn't look like himself."
They can then live outside the ship. Then a female alien comes up to the ship and the women on the ship have to decide whether to let her on or not.
Max called for attention. "There are nine Meads outside who want to come in. They have different names, but they're all Pat Mead."
They frowned or looked blank, and George Barton asked, "Why didn't you just let them in? I don't see any problem."
"One of them," said Max soberly, "is a girl. Patricia Mead. The girl wants to come in."
There was a long silence while the implication settled to the fear center of the women's minds. Shelia the beautiful felt it first. She cried, "No! Please don't let her in!" There was real fright in her tone, and the women caught it quickly.
Elsie clung to Jerry, begging, "You don't want me to change, do you, Jerry? You like me the way I am! Tell me you do!"
The other girls backed away. It was illogical, but it was human. June felt terror rising in herself. She held up her hand for quiet and presented the necessity to the group.
"Only half of us can leave Minos," she said. "The men cannot eat ship food; they've been conditioned to this planet. We women can go, but we would have to go without our men. We can't go outside without contagion, and we can't spend the rest of our lives in quarantine inside the ship. George Barton is right—there is no problem."