* Common Sense: The idea that people will resort to extreme measures when they feel they have nothing left to lose is deeply rooted in human experience. This sentiment likely predates any specific recorded instance of the saying.
* Proverbs: Many cultures have proverbs expressing the same basic concept. For example, "A drowning man will clutch at a straw" is an old English proverb, while "When the well is dry, we know the worth of water" is a Chinese proverb.
* Literature: The concept is found in literature throughout history. For instance, Shakespeare's character Macbeth says, "The last [of all the crimes] is done to cure the first." This reflects the idea that desperate actions can be a response to earlier, desperate situations.
* Modern Usage: The exact phrase "Desperate people do desperate things" likely gained popularity in the 20th century, becoming a commonplace expression in English.
In short: The saying is a distillation of a universal human understanding about the lengths to which we may go when we feel cornered. It's not a specific quote with a clear origin, but rather a reflection of a deeply ingrained human insight.