What is the origin of the phrase 'Drastic times call for drastic measures'?

The saying is actually "Desperate times call for desperate measures."

The origin of this proverb:

This is a variant of the proverb "Desperate diseases must have desperate remedies." This goes back to the Latin _extremis malis extrema remedia_ 'extreme remedies for extreme ills.'

The earliest English version given by the Concise Oxford Dictionary of Proverbs is the following:

1539 R. Taverner tr. _Erasmus' Adages_ 4 A stronge disease requyreth a stronge medicine.
(Thanks to Fred Shapiro, Yale Law Library)