Consider the quality of holograms these days. Vic Fontaine, for example, can perform the job of therapist better than the station's actual therapist (the one who has multiple lifetimes of life-experience). Voyager's holographic doctor is just as good as a flesh-and-blood doctor, in addition to being an accomplished singer, and even capable of taking other roles on the ship.
These holograms are run on the ship's computers, which tells us that the ship's computers have more than enough AI capabilities to do the jobs that the holograms manage to do. In other words, there's no reason a ship couldn't perform its mission completely autonomously, without a living crew.
And yet, we still have ships full of officers going on missions to explore the galaxy, fight wars, etc. etc. Why are they there at all? They're there because they don't have anything better to do with their time. Nobody needs to earn money anymore -- most of life's problems have been solved. Aside from a very small number of particularly creative / intelligent people, there are few things a human could do that couldn't be done just as well (or better) by a computer/robot/hologram.
So with very few exceptions, Starfleet crews are just dressing in costumes and pretending to be useful parts of the ship's mission. Some crewmembers realize this; most don't. (I think Riker was starting to figure it out after taking a holodeck adventure as a member of the NX-01's crew.)
Who do you suppose the last "real" crews were before the ships' AIs took over? Why do you suppose the ships' AIs are letting the humanoid crews tag along?