A person from Kentucky was called a “Red Horse” in the 19th century. The red-tail sucker, a fish that was also called “red horse,” was common in the Ohio River. “The spokesman was evidently a ‘red-horse’ from Kentucky” was written in a December 26, 1833 letter by New York editor and poet Charles Fenno Hoffman (1806-1884) that was published in the New-York (NY) American newspaper on February 22, 1834. The nickname is mostly of historical interest today.
 
Another 1830s nickname for a Kentuckian was “Corn-cracker.”
   
 
(Oxford English Dictionary)
red horse, n.
Any of various North American suckers (freshwater fishes of the family Catostomidae), of the genus Moxostoma or related genera, having the lower fins (and often the dorsal fin and tail) bright red or orange. Freq. with distinguishing word. Also more fully red horse fish, red horse sucker.
(...)
U.S. A nickname for: a native of Kentucky. Now rare (hist. in later use).
1833   C. F. Hoffman Let. 26 Dec. in N.-Y. Amer. (1834) 22 Feb. 2/1   The spokesman was evidently a ‘red-horse’ from Kentucky.
1838   T. C. Haliburton Clockmaker 2nd Ser. xix. 289   These last have all nicknames. There’s the hoosiers of Indiana,..the red horses of Kentucky.
1859   in L. R. Hafen & A. W. Hafen Rep. from Colorado (1961) 17   Here they come..from every place—Hoosiers, Suckers, Corn crackers, Buckeyes, Red-horses.
     
Google Books
June 1820, The Western Review and Miscellaneous Magazine, “Fishes of the River Ohio” by C. S. Rafinesque, pg. 365:
RED-TAIL SUCKER. Catostomus erythrurus.
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A fine species, not uncommon in the Ohio, Kentucky Cumberland, Tennessee, &c. Vulgar names Red-horse, Red-tail, Horse-fish, Horse Sucker, &c.
 
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22 February 1834, New-York (NY) American, “Review of the Week,” pg. 2, col. 1:
There was a long-haired “hooshier” from Indiana, a couple of smart-looking “suckers” from the southern part of Illinois, a keen-eyed leather-belted “badger” from the mines of Ouisconsin, and a sturdy yeomanlike fellow, whose white capote, Indian mockasons, and red sash proclaimed, while he boasted a three years residence, the genuine wolverine, or naturalized Michiganian. Could one refuse a drink with such a company? The spokesman was evidently a “red-horse” from Kentucky, and nothing was wanting but a “buckeye” from Ohio, to render the assemblage as complete as it was select.
 
Google Books
Ralph Doughby’s Esq.
Volume 3
By Charles Sealsfield
Zürich: Orell, Fußli & Co.
1835
Pp. 12:
Suckers von Illinois, und Badgers von den Bleiminen Missouris, und Wolverines von Michigan, und Buckeyes von Ohio, untermengt mit Red-horses vom alten Kentucky, ...
 
6 November 1835, Western Constellation, “Nick Names,” pg. 2, col. 6:
Thus, the people of Indiana are called Hoosiers; of Illinois, Suckers; of Missouri, Pukes; of Ohio, Buckeyes; of Kentucky, Red Horses; of Tennessee, Medheads; of Michigan, Wolverines—Yankees are called Eels, and Virginians Corncrackers. (...)—New Yorker.
 
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The Clockmaker; or The Sayings and Doings of Samuel Slick, of Slickville
By Thomas Chandler Haliburton
London: Richard Bentley
1838
Pg. 289:
These last have all nicknames. There’s the hoosiers of Indiana, the suckers of Illinoy, the pukes of Missuri, the buckeys of Ohio, the red horses of Kentucky, the mud- heads of Tenessee, the wolverines of Michigan, the eels of New England, and the corn-crackers of Virginia.
 
Google Books
8 September 1838, New-York (NY) Mirror (New York, NY), pg. 86, col. 2:
These last have all nicknames. There’s the Hoosiers of Indiana, the Suckers of Illinoy, the Pukes of Missouri, the Buckeyes of Ohio, the Red Horses of Kentucky, the Mud-heads of Tennessee, the Wolverines of Michigan, the Eels of New-England and the Corn-crackers of Virginia.
 
Google Books
The Attaché; or, Sam Slick in England
By Thomas Chandler Haliburton
London: Richard Bentley
1843
Pp. 236-237:
Why, as I am the livin’ sinner that’s the Hoosier of Indiana, of the Sucker of Illinois, or the Puke of Missouri, or the Bucky of Ohio, or the Red Horse of Kentucky, or the Mudhead of Tennesee, or the Wolverine of Michigan or the Eel of New England, or the Corn Cracker of Virginia!
 
Google Books
Life in the New World,
Or, Sketches of American Society
By Charles Seatsfield
New York, NY: J. WInchester
1844
Pg. 55:
They would begin the struggle rather to-day than to-morrow; the Hooskiers from Indiana, the Suckers from Illinois, the Pukes from Missouri, the Red-horses from Kentucky, the Buckeyes from Ohio, the Wolverines from Michigan, the Eels from New England, the Mudheads from Gennessee, the Corncrackers from Virginia, they are all ready.
 
Google Books
Dictionary of Americanisms (Second Edition)
By John Russell Bartlett
Boston, MA: Little, Brown and Company
1859
Pg. 358:
RED HORSES. A nickname applied to the natives of Kentucky.
 
Google Books
U. S.
An Index to the United States of America
Compiled by Malcolm Townsend
Boston, MA: D. Lothrop Company
1890
Pg. 76:
NICKNAMES APPLIED TO THE PEOPLE OF THE STATES.
(...)
Kentucky…Red Horses...Bartlett notes “a nickname given to natives of Kentucky,” but gives no reason.
 
Google Books
Encyclopedia of Kentucky
By Nancy Capace
St. Clair Shores, MI: Somerset Publishers, Inc.
1999
Pg. 3:
The sobriquet Red Horses, given to the people of Kentucky, probably alludes to a kind of fish, a large red sucker (Castostomus dequesnii) commonly found in the Ohio River and its tributaries.