Douglas Harper's Etymology Dictionary
worry wart
1956, from comic strip "Out Our Way" by U.S. cartoonist J.R. Williams (1888-1957). According to those familiar with the strip, Worry Wart was the name of a character who caused others to worry, which is the inverse of the current colloquial meaning.
Wiktionary
worry wart
alt. (context idiomatic English) One who worries excessively or unnecessarily. n. (context idiomatic English) One who worries excessively or unnecessarily.