One of the obviously ridiculous aspects of Star Trek and similar science fiction shows is that humans not only make the larger, more strategic decisions ("Attack that ship!"), they also make more detailed decisions that would certainly be automated. There is absolutely no way, for example, that if sensors detected a vessel of unknown intentions charging their weapons the procedure would be for the weapons officer to notice this, speak to the captain, after which the captain considers the situation, eventually realizing that if shields are not raised the ship will be destroyed, so he tells the weapons officer to raise shields, which he then does. No way! The shields would be raised automatically. Depending on pre-programmed decisions depending for situations matching those encountered, an automatic attack (or retreat) would probably also be launched, though the captain could disable this feature when it was deemed in appropriate.
In fact, most of the bridge activity makes no sense whatsoever; it would mostly be replaced by automation except for the big decisions that (to be meaningful) have to be made by the humans (or Vulcans, or whatever): things like, "Are we friends or are we enemies?"
Engineering would not be much better. A sailing ship of the 18th century could make most repairs using tools and materials already aboard ship, possibly with the use of raw materials (like trees) that could be obtained without further help from civilization. If a circuit board on a modern aircraft carrier is fried, though, the ship does not have the facilities to fabricate a replacement. The more technically sophisticated a ship is, the more dependent it is on a technically sophisticated society to supply its components. If the Enterprise's engines were working properly, they probably would need to be controlled automatically (human reaction time again being too slow); if the engines were broken, it would probably be impossible to fix them without a tow to a shipyard.
What is interesting, though, is that increasing automation comes at a cost, the automation paradox. As the crash of Air France flight 447 makes clear, we have already reached the point where that paradox has begun to be a problem. If that is true on a modern jetliner, it will be vastly more true on future spaceships.
It would be interesting to see this dealt with seriously in a science fiction context. Maybe -- just maybe -- it could be used to justify having a helmsman, a weapons officer, etc., rather than automating the whole operation.