This is a fantastic question. I wrote my undergraduate thesis on the 4th crusade, so I'll give this a shot. This information is largely coming from Christopher Tyerman's "God's War." If you're interested, you could also check out the written account of Nicetas Choniates, the head of Alexius III's civil service. It's been years since I've read through his specifically, but it's possible he might have discussed the lack of a functioning navy by the time the Crusaders reached Constantinople.

The TL;DR answer to your question is that the navy essentially fell out of use around the time Alexius II succeeded Manuel I. Alexius II was a minor when he assumed power, and the relatively prosperous political arena began to devolve into "factionalism and chaos." (God's War, pg. 536). However, even during the reign of Manuel I, the empire made a series of poor military and defense decisions. During this time, the empire began subcontracting military functions to foreign mercenaries. This transition from domestic military reliance to reliance on outside forces eventually proved ruinous when Alexius II came to power. According to Tyerman, "Between 1180 and 1204, fifty-eight coups, rebellions and conspiracies against the emperor [occurred], at least five successful, in 1182-3, 1185, 1195, 1203 and 1204." Id. This instability, coupled with persistent foreign threats the empire had faced even prior to Manuel I's rule, crippled the Byzantine economy. This, in turn, led to further weakening of the navy. This situation had not improved by the time Alexius III assumed power, and it wasn't until the Crusaders neared Constantinople in 1203 that Alexius III realized that the once great Byzantine navy consisted of twenty "rotting and worm-eaten small skiffs." Id.

This, of course, is a very condensed version of the history of the degradation of the Byzantine navy. Tyerman discusses it extensively in the book I cited to above, and his account is definitely worth reading!