“Nutty” has been used as slang for “crazy” since at least the 1890s. Fruitcakes usually contain nuts. “At this point I realized that Henry had become as nutty as a fruit cake” was cited in print in 1913. “Powers is nutty as a fruitcake” was cited in 1914.
 
   
The Free Dictionary
nutty as a fruitcake and nuttier than a fruitcake
crazy. (*Also: as ~.) Don’t pay any attention to John; he’s nutty as a fruitcake. Mary’s schemes for making money are nuttier than a fruitcake.
See also: fruitcake, nutty
   
(Historical Dictionary of American Slang)
nutty adj.
crazy; eccentric, nuts.
1912-14 in E. O’Neill Lost Plays 171: We sure are as nutty as a fruitcake or we wouldn’t be here.
1935 Lorimer & Lorimer Heart Specialist 163: You’re as nutty as a fruitcake.
1945 Hartman & Shavelson Wonder Man (film): You’re nutty as a fruitcake!
1958 “R. Traver” Murder 223: Nuttier than the proverbial fruitcake.
 
(Oxford English Dictionary)
nutty, adj.
colloq. (orig. U.S.). Deranged, not right in the head; crazy; mad, insane. Also in weakened sense: eccentric; wildly enthusiastic or passionate (about something or someone).
1892 W. NORR Stories Chinatown 7 The boys guyed him..and volunteered the information that he was either ‘nutty’ or ‘off his feed’.
1898 S. CRANE in Cosmopolitan Dec. 169/1 ‘What’s the matter with that feller?’ asked Martin. ‘Nutty,’ said the man.
1918 Stars & Stripes 19 Apr. 2/3 ‘Gosh,’ said the Indiana private, ‘that bread was as black as a derby.’ ‘You’re nutty,’ said another. ‘It was blacker than that; didn’t you ever see a brown derby?’
 
Chronicling America
5 June 1913, The Evening World (New York, NY), pg. 16, col. 2:
At this point I realized that Henry had become as nutty as a fruit cake.
 
Chronicling America
10 June 1914, Norwich (CT) Bulletin, “Coulon Loses Bantamweight Title,” pg. 3, col. 3:
Says Bridgeport Farmer: Bugs Raymond and Rube Waddell had nothing on Bill Powers when it comes to leading the squirrels a merry chase. Powers is nutty as a fruitcake.
 
Google Books
The ghost story: a one-act play for persons of no great age
By Booth Tarkington
Boston, MA: Baker’s Plays
1922
Pg. 47:
And nuttier than a Christmas fruit cake! In fact, they are all nuts!
     
7 December 1924, Chicago (IL) Daily Tribune, pg. G2:
“Nutty as a fruit cake.”
     
Google Books
Alias Ben Alibi
By Irvin Shrewsbury Cobb
New York, NY: Grosset & Dunlap
1925
Pg. 40:
“Nutty as a fruit cake, the way I figure him,” said Sheridan.
 
Google Books
“Here comes the bride—,”: and so forth
By Irvin Shrewsbury Cobb
New York, NY: George H. Doran Company
1925
Pg. 252:
...and assuming that the defendant’s Aunt Jane, in Wilkes-Barre, was as nutty as a fruitcake,...
   
Google Books
Round trip: a novel
By Don Tracy
New York, NY: Vanguard Press
1934
Pg. 3:
He’s nuttier than a fruit cake, usually, but a good guy.
   
6 May 1934, New York (NY) Times, “Miss Allen and Mr. Burns at Home,” pg. X4:
“They’re convinced she s nutty as a fruit cake.”
(George Burns on Gracie Allen—ed.)
   
New York (NY) Times
Movie Review
The Good Fairy
THE SCREEN; The Radio City Music Hall Presents a Screen Version of Molnar’s Fantastic Comedy, ‘The Good Fairy.’
By ANDRE SENNWALD.
Published: February 1, 1935
(...)
It is, to borrow a sacred Burns and Allen simile, as nutty as a fruit cake, and it contains some of the most painfully hilarious merriment of the new year, if you except W. C. Field’s Micawber in “David Copperfield.”
 
Bye Bye Texan-ese
By L. E. Guillot
Dallas, Texas (published by author)
1962
Pg. 28:
Nutty as a fruitcake