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Theodore Sturgeon's story "The Golden Helix" was initially published in June, 1954. Sturgeon later introduced his 1979 anthology of the same name with a preface stating that he wrote the story in 1953, "a good span of years before" the double helix for genes was discovered, however the scientific paper that introduced the double helix to the world came out in April, 1953.

"Far more remarkable, to me, than any other aspect of the intricate
plot of this story is the fact that it was written in 1953, a good
span of years before the double spiral of the DNA molecule was
discovered, with its astonishing role in evolutionary structures. This
makes the story a sort of quasi-mystical precognition - something I
was not and could not be aware of when I wrote it. This is by no means
the only time this has happened. Well after the fact, readers have
unearthed in my work devices, events, or phenomena that I couldn't
possibly have known of at the time I wrote them: Velcro, illuminated
watch-dials, certain breakthroughs in cancer research, automobile smog
devices, and a good many other things. The average gap between these
appearances in my typewriter and the emergence of them in the real
world seems to be about fifteen years. I claim no special superiority
for this, and admit to a good deal of humility. There are times when I
feel like no more than a length of pipe, through which Something pours
these things into my manuscripts."

Has anybody examined the chronology to see if Sturgeon had a hint of the double helix before he wrote "The Golden Helix"?

The correct double helix structure of DNA was published by Watson and Crick in April 1953. However, the idea that DNA molecules coiled into helices was somewhat older. Before the precise x-ray diffraction data that Rosalind Franklin produced in 1952–1953 was available, cruder measurements had already indicated that DNA molecules formed something like a solid cylinder form.

On the basis of this, several people tried to devise helical structures for the molecule. Watson and Crick had previously proposed a triple helix structure in 1951, but after getting Franklin's input, they realized that it would not hold together properly. The next year, Linus Pauling, who had already worked out the structures of α-helices and β-sheets in proteins, came up with a similar erroneous structure. The story of Pauling's work is given here:

On November 25, 1952, three months after returning from England, Pauling finally made a serious stab at a structure for DNA. The immediate spur was a Caltech biology seminar given by Robley Williams, a Berkeley professor who had done some amazing work with an electron microscope. Through a complicated technique he was able to get images of incredibly small biological structures. Pauling was spellbound. One of Williams's photos showed long, tangled strands of sodium ribonucleate, the salt of a form of nucleic acid, shaded so that three-dimensional details could be seen. To Pauling the strands appeared cylindrical. He guessed then, looking at these black-and-white slides in the darkened seminar room, that DNA was likely to be a helix. No other conformation would fit both Astbury's x-ray patterns of the molecule and the photos he was seeing.

Even better, Williams was able to estimate the sizes of structures on his photos, and his work showed that each strand was about 15 angstroms across. Pauling was interested enough to ask him to repeat the figure, which Williams qualified by noting the difficulty he had in making precise measurements.

The next day, Pauling sat at his desk with a pencil, a sheaf of paper, and a slide rule. New data that summer from Alexander Todd's laboratory had confirmed the linkage points between the sugars and phosphates in DNA; other work showed where they connected to the bases. Pauling was already convinced from his earlier work that the various-sized bases had to be on the outside of the molecule; the phosphates, on the inside. Now he knew that the molecule was probably helical. These were his starting points for a preliminary look at DNA. He still lacked critical data - he had no decent x-ray images, for instance, and no firm structural data on the precise sizes and bonding angles of the base-sugar-phosphate building blocks of DNA - but he went with what he had.

It was a mistake. After a few pages of theorizing, using sketchy and sometimes incorrect data, Pauling became convinced — as Watson and Crick had been at first — that DNA was a three-stranded structure with the phosphates on the inside. Unfortunately, he had no Rosalind Franklin to set him right.

The upshot of all this is that by the time Sturgeon wrote his story, it was reasonably well known (among biochemists, at least) that DNA had some kind of helix structure. So the author could certainly have picked up this information somewhere, even if he had not yet learned about Watson and Crick's correct double helix structure.