The ball got rolling, as Pronunciation of Ancient Greek in teaching – Wikipedia notes, in the early Renaissance, a generation before Erasmus. Erasmus published the system that prevailed in the West since, and that was a closer approximation of the modern reconstruction than Modern Greek pronunciation was:
The study of Greek in the West expanded considerably during the Renaissance, in particular after the fall of Constantinople in 1453, when many Byzantine Greek scholars came to western Europe. At this time, Greek texts were universally pronounced using the medieval pronunciation which survives intact to the present day.
From about 1486, various scholars (notably Antonio of Lebrixa, Girolamo Aleandro, and Aldus Manutius) judged that this pronunciation appeared to be inconsistent with the descriptions handed down by ancient grammarians, and suggested alternative pronunciations. This work culminated in Desiderius Erasmus’ dialogue De recta Latini Graecique sermonis pronuntiatione (1528). The system propounded in this work is called the Erasmian pronunciation.
The pronunciation described by Erasmus is very similar to that currently regarded by most authorities as the authentic pronunciation of Classical Greek (notably the Attic dialect of the 5th century BC). However Erasmus did not actually use this pronunciation himself.
The Modern reconstruction was informed by more close reading of the ancient authorities, internal reconstruction, better knowledge of ancient dialects through inscriptions, and comparative historical linguistics. Once you look at the comparisons of Greek with Sanskrit and Latin, the reconstruction becomes pretty obvious.