Question

What event was the most significant in damaging the strength of the Byzantine Empire during the crusades?

Answer 1

Probably the sacking of Constantinople in the Fourth Crusade. The short answer is that the Crusaders needed to pay for ships to take them to Egypt and, convinced by the Venicians (who were economic rivals of the Byzantines), sacked Constantinople and set up a short lived Latin Empire in the region with a puppet emperor as a result. The Byzantine Empire was already on the decline by the time that happened in the early 13th century making this a lot like piling on the damage that was already there.