"She drinks a glass of wine."
Translation:Ŝi trinkas glason da vino.
Because da is a preposition, and like most prepositions, is followed by the nominative case. (The exceptions I can think of are those that can indicate location or destination of movement, and take different cases depending on which of those they signify.)
A bit like how in English you say "They drink water" but "Both of them drink water" -- you would not say "Both of they drink water" because the form "them" is demanded by the preposition "of" rather than by the fact that "both of them" is the subject.
So here, too, you need vino because it's the complement of the preposition da, regardless of whether the prepositional phrase da vino is part of the subject or the object.