According to J.E. Lighter, Random House Historical Dictionary of American Slang (1993), crust in the relevant sense goes back at least to 1900:
crust n. effrontery. [First four cited examples:] 1900 D[ialect] N[otes] II 31: Crust n. Forwardness. 1908 in H.C. Fisher A. Mutt 72: You've got a crust to speak to me without an introduction. 1915 D[ialect] N[otes] IV 233: That guy certainly has some crust. 1929 in Galewitz Great Comics 117: Say, Emmy you gotta a lotta crust blabbin' to Mamie that you heard Uncle Willie talkin' about bein' out woth Mrs. Smokehouse on the Coast. [Last cited example:] 1988 N.Y. Newsday (July 6) 11: Pie-tosser has dropped routine, but crust remains.
The earliest instance that Lighter cites comes from an issue of Dialect Notes containing Eugene Babbitt, "College Words and Phrases" (1900), which further specifies that crust in the sense of "forwardness" was recorded at Brown University (in Providence, Rhode Island) and Wesleyan University (in Middletown, Connecticut) in 1900 or slightly before.
Recent instances of the expression in print seem to be quite rare—and I don't think I've ever heard anyone use the expression in real life. But I did find an instance of it in a fairly recent novel. From Charlotte Armstrong, Mischief (2012):
Miss Ballew bridled but stood her ground. "Snooping or not, I wish to see the child."
"See her?" For the first time, Miss Ballew felt that her words were heeded.
"Yes, see her for myself."
"You've got a crust!"
"Nevertheless, if I do not see her, I intend to call the authorities." So much for rudeness, Miss Ballew's eyebrows remarked.
The historical setting for this novel is unclear from the chapters excerpted in Google Books, but I can't confidently say that it is supposed to be contemporary. Still, if people writing books are still using crust to mean effrontery it is certainly possible that some people in the real world are, too.