The Commodore, a filibuster steamer out of Jacksonville, sank off the Florida coast early on 2 January 1897. Four men got away in the ten-foot dinghy: the captain, the cook, an oiler named Billie Higgins, and Crane, who had signed on as a seaman and was a newspaper correspondent. They spent that day and the night after it at sea. In the morning they ran the boat through the surf near Daytona. Cyrus Day, who checked the story against the record in 1957, puts the landing "between 7:30 and 10:00 a.m. on Sunday, January 3". Higgins drowned in the surf.
Crane wrote it up twice. The first version was a newspaper dispatch, "Stephen Crane's Own Story", datelined Jacksonville, 6 January. The second was the story, in Scribner's Magazine that June. The 1898 London edition, which is the text I read, gives it the subtitle "A Tale intended to be after the Fact". The story's last paragraph:
When it came night, the white waves paced to and fro in the moonlight, and the wind brought the sound of the great sea's voice to the men on shore, and they felt that they could then be interpreters.
"When it came night" reads as the evening of the landing, Sunday 3 January. The new moon fell that same day, at 06:03 Universal Time (NASA's eclipse site, "Phases of the Moon: 1801 to 1900"). At Daytona that was about 00:40 local time, in the middle of the night the men spent in the boat.
I ran the sky for Daytona with PyEphem. On the night in the boat the moon was below the horizon from late afternoon until after sunrise. On the evening of the landing it was a crescent under one percent lit, and it set about 34 minutes after the sun. Nobody would have seen it, and it threw no light. The next evening it was three and a half percent lit and set before seven. There was no moonlight on the beach that night, and nothing worth the name for several nights after.
The dark the story keeps
The night in the boat is written without a moon, and nothing in it needs one. As the light goes, "a pale star appeared, just lifting from the sea", and it is the lighthouse. Then:
As darkness settled finally, the shine of the light, lifting from the sea in the south, changed to full gold. On the northern horizon a new light appeared, a small bluish gleam on the edge of the waters. These two lights were the furniture of the world. Otherwise there was nothing but waves.
Two lights make up the furniture of the world. The shark that circles the boat that night is never seen as a shape: "a gleaming trail of phosphorescence, like blue flame, was furrowed on the black waters", then "the long sparkling streak" and "the whiroo of the dark fin". When the correspondent wants something to plead with, what he feels nature says to him is "A high cold star on a winter's night". When he wakes, "the sea and the sky were each of the grey hue of the dawning". For a whole night at sea there is no moonlight in the story, and none in the sky either.
The dispatch has no moon at all. I searched the word inside the 2000 Frus and Corkin edition, which reprints the dispatch at pages 337 to 345. "Moon" turns up eleven times in the volume, none of them in those pages. The only "Open Boat" hit is the last sentence.
The fact-checker's sky
Day's article, "Stephen Crane and the Ten-Foot Dinghy" (Boston University Studies in English 3, 1957), is the best-known check of the story against the facts of the wreck. He used the weather records. He reports temperatures at New Smyrna and Jacksonville for 2 and 3 January, the wind, and the sky: at New Smyrna at seven on the morning of the 3rd, "the skies were clear". HathiTrust's full-text index of the volume finds "moon" and "moonlight" only on pages 14 to 21, in an essay on Fitzgerald, and nowhere in Day's pages. Google Books agrees on every page it will show. I can search that text but not read it, so all I can say is that the word isn't there.
So the night in the boat is accurate, maybe more accurate than Crane knew. It was the darkest night of the month, and he never put a moon in it. The night on shore is where the story and the sky part.
After the fact
I don't think it's a slip worth correcting. Look at the sentence it sits in. Through the whole story the men in the boat can only take the sea as it comes. Its first line is "None of them knew the colour of the sky". In the last line they are on land, the oiler is dead, and "they felt that they could then be interpreters". Interpreting is what the story does from its subtitle on, after the Fact. The record could not have shown Crane moonlight on the white waves that night. He gave it to them anyway, in the same sentence that tells you the men have started to interpret.
I haven't found anyone who has noticed the date of the new moon. That isn't the same as nobody having noticed. My checks were a dozen Google Books searches, two Google searches, a HathiTrust full-text search, and the two indexes of Day. Day's isn't the only study of the wreck, and I'd expect the moon to turn up somewhere in Crane scholarship I couldn't search.
Sources: Crane, "The Open Boat" (Project Gutenberg #45524, from The Open Boat and Other Stories, Heinemann 1898); "Stephen Crane's Own Story", in The Red Badge of Courage; Maggie, a Girl of the Streets; and Other Selected Writings, ed. Frus and Corkin (Houghton Mifflin, 2000), pp. 337-45, searched at archive.org; Cyrus Day, "Stephen Crane and the Ten-Foot Dinghy", Boston University Studies in English 3 (1957), pp. 193-213, snippet and full-text search only (landing time p. 194, weather p. 200); NASA GSFC eclipse site, "Phases of the Moon: 1801 to 1900"; PyEphem for Daytona, 29.21 N, 81.02 W.