I commented on this in the Badly Designed thread, saying, "really think a crew of 3000, plus a smattering of civilians, would make more sense. The ship is huge; and having it run by 200-300 people at any given time doesn't seem sufficient given all the things that need to be done -- operations, maintenance, research, various support missions in local systems, continuing education, weekends/days off (I know ship duty is usually 7 days, but this is a multi-year journey), and the list goes on."
To the point about automation and computers... One of the novels -- I forget which one off hand -- addressed this, and I really love the line of thinking. Lt Uhura was having a conversation with some starbase doctor who was being a bit of a snob regarding the Enterprise's low-tech setup with physical switches, wires, and circuit boards in Sickbay. She told him that in an emergency or combat situation, everything needed to be easily accessible and repairable even without sophisticated tools. Occam's Razor. You can replace a switch or bypass a circuit in seconds. That's a lot harder to do with a busted touchscreen.
Applied to the crew size, it seems to me that the automation argument holds water, but would carry with it all kinds of dangers. In an emergency, sophisticated tech is hard to repair. In combat, 1000 people -- many of whom are not security-focused, and some of whom are civilians -- defending 200 acres seems like a bit of a challenge. Etc.