Well, it was mostly a process of winding down and gradual losses of colonies as the Venetians dealt with the consolidation of the Ottoman and Habsburg Empires around them. If you want an arbitrary date, 1669 and the loss of Crete to the Turks marked the point where it started to become uncouth in Venice. Crete was the main illicit center for the slave trade after the 1381 laws, and the main sugar colony that necessitated needing slave labour. After the capture of Crete, slavers taking advantage of the weakened naval power of the Venetians would sometimes have intermediaries sail ships of captured Christians right into Venetian towns to beg people passing by to pay their ransom, and in the last 200 years of Venice, it became such a large and expensive problem that churches asked for alms and government bureaucracies set up organized programs to pay certain ransoms in hopes of getting slaves back before they had to convert to Islam. Sometimes slavers would have captured Christians write dramatic letters home to make their plight in Tunis or other Barbary slave centers look terrible, which made the whole idea of the institution unpopular with most of the Venetian public.

If you mean when was slavery actually wiped out entirely it wasn't until Napoleon's conquests.