What action does Barton take at the story’s climax?

Quite simply, the text tells us that Barton releases Marilyn into space at the story's climax. The main conflict of the story (whether Marilyn will live or die) provides suspense.
As for Barton, he is shown to be reluctant in the performance of his macabre duty. In the story, he is an EDS pilot, faced with a difficult problem that tests his humanity to its limits. When he discovers Marilyn, he is ambivalent about releasing her into space. If she had been a fugitive from justice or an avaricious opportunist (these were the usual stowaways he came across in his line of work), he would have had no problem in immediately ejecting the stowaway into space.
However, Marilyn is a friendly, pretty, and unassuming teenager. Her only reason for boarding the Stardust as a stowaway is to see her brother, Gerry, whom she has not seen for a very long time. Marilyn tells Barton that she and Gerry are the only children their parents have. This makes Barton's task even more difficult.
In the end, Barton has little choice (according to the story, anyway) but to order Marilyn into the airlock and to eject her into space.