Why didn't the therapist ever open the envelope at the end of The Departed? Whose baby, if any, was it?

Answer 1

Just because we didn't see her open it doesn't mean she didn't. She was supposed to open it if something "happened' to Costigan (DiCaprio's character), and the action quickly skipped from that final elevator scene with Costigan to his funeral. Presumably she opened the envelope sometime after his death, and she may even have been instrumental in contacting Dignam (Mark Wahlberg) afterward. Sometimes the dots are left to the audience to connect, and it doesn't matter too much how we connect them.

One thing to note is that in the movie The Departed was based on, the Hong Kong film Infernal Affairs, the Matt Damon character gets away with it at the end.

As for whose baby it was--I guess it was irrelevant after all that happened; I had always assumed that it was Sullivan's (Matt Damon).