Etymology of "corny"
Its meaning as old-fashioned is from the 30's and appears to come from jazz:
Corny:
The adjective “corny” has a shorter history. It’s been a term of derision only since the 1930s, when something that was “corny” or “cornfed” or “on the cob” was rustic, countrified, old-fashioned, or behind the times – and hence trite or hackneyed.
It first was used by jazz musicians, who called a style of playing “corny” if it was outmoded or worn out. Here’s the OED‘s first citation, from 1932: “The ‘bounce’ of the brass section … has degenerated into a definitely ‘corny’ and staccato style of playing.” (Imagine a rube fresh from the cornfields trying to make a splash in the big city and you’ll get the idea.)
(grammarphobia.com)
The Wold Detective, besides the more generally accepted origin from jazz, offers another possible etymology from old 'seeds catalogues'.
Some of the earliest documented uses of "corny" were among jazz musicians in the late 1920s, who used the term to mean an old-fashioned or trite style of jazz, likening it to music that might be heard in the boondocks, perhaps at a square dance. Derogatory references to rural inhabitants, culture and customs as being crude and unsophisticated were nothing new at the time, of course, and persist to this day in such terms as "hick" and "rube" (short for Henry and Reuben, once considered typical "country" names), as well as such slurs as "hayseed." For a jazz musician whose status depended on constantly forging new styles of music, nothing was more deadly than being considered "corny," and by the 1930s the term had percolated into general usage in the sense of "trite or sentimental."
Another possible origin of "corny" casts rural folk in a considerably better light. Seed catalogs in the late 19th and early 20th centuries, it is said, often contained humorous stories and jokes interspersed among the product listings. As the jokes tended to be unsophisticated and obvious, the genre came to be known among farmers as "corn jokes," or, eventually, simply "corny."
It is possible, of course, that both theories are true. Perhaps it was farmers who first came up with "corny," only to have the term turned against them by urban sophisticates.
Here's a theory:
1946 M. Sandoz in American Speech (American Dialect Society) 21 234/1 "The seed catalog [from c 1890 to 1910]..featured a great variety of seed corn..interspersed with short jokes and riddles, sometimes even cartoons. The jokes were all time-worn and over-obvious and were called corn catalog jokes or corn jokes, and any quip or joke of that nature was called corny."