Prometheus Bound is a drama: it is a literary work in which the words are assigned to characters who are supposed to speak them onstage. That makes it a drama.
Fair enough. Ancient Greek theater had two kinds of plays: comedy and tragedy. Take a guess which one this is. (You have a 50% chance of being right.)
Yep, it's a tragedy all the way. Serious or somber theme? Well, how about rebellion against the gods? Main character is a great person? Check—Prometheus is a Titan. This great person is destined to be destroyed through a flaw of character or conflict with some overpowering force? (Does pig-headedness count as a character flaw?)
Bring out the tissues.