What kind of impact did the short story have on?
The protagonist of James Thurber's story "The Secret Life of Walter Mitty" has lent his name to the type of ineffectual, self-glorifying daydreaming that the short story revolves around, in the same way that the Marquis de Sade's name was borrowed to form the word "sadism." References to "Walter Mitty syndrome" can be found in such works as the Handbook of Psychological Assessment by Gary Groth-Marnat (2003). It might be argued that now we have the term "Walter Mitty syndrome" as an "official" diagnostic label, it has become easier to identify and discuss the associated behavior.
One example of the use of "Walter Mitty" as a slang phrase is found in the British military, to denote and dismiss applicants to the elite SAS who are motivated primarily by personal vanity. British military slang also includes the "Walt" (a person fantasizing being a soldier, or being a soldier in a more specialized or prestigious branch than really occupied) and the "Weekend Walt" (a reservist who is lost in similar fantasies).