In the opening credits of Star Trek TNG, I noticed the captions have the line as “These are the voyages of the Starship Enterprise…” But to my ears, it sounds like Picard says “…a Starship Enterprise”
Does he always say it that way? What about when Picard says the line at the end of the episode “These are the Voyages”?
He says “the”. It’s pretty clipped, since he’s English and speaks with an upper class accent. If you don’t hear the “th” because the sound is down, or because the ambient noise is up, you’ll think he said “a”.
Unlike Neil Armstrong, who definitely said neither. 
I’ve always heard it as “the.” The introduction is the same as that as the original Star Trek series, except with “its five-year mission” changed to “its continuing mission” and “where no man has gone before” changed to the more politically correct “where no one has gone before.”