Depends on where the characters are at any particular moment. I mean, outer space is just about as dark as you can get, but I'd think the bridge should be quite well-lit.
I think at best it shouldn't be an overall bleak and dreary show (it should be a show that people would like to be on, IMO). It should have both some very light episodes and also some very dark (if not outright scary) episodes.
It should know when to be dark and when to lighthearted and fun. If it's all dark, all the time, then people are probably going to tune out. Nothing makes me stop watching a show quite like likable characters getting screwed all the time for no real reason beyond drama.
Too be fair, there a lot of dark shows on TV that people like and get a lot of critical praise, but Star Trek should not be as dark as Game of Thrones or Breaking Bad. I think a new show will at least have to get with the with the times a bit. Less technobabble and fewer morality plays where the situation is black and white and the "bad side" is represented by a silly cartooinsh villain, either have moral dilemmas that actually gray or if it is black and white, show why something is bad in all its ugliness. Chain of Command showed torture in all its ugliness and that was effective. Also better continuity through out the show would be good.
Babylon 5, Deep Space 9, Farscape, Whedon's shows, they all got at least close to as Dark as NuBSG. Current shows Falling Skies and The Walking Dead (I don't know the Pay Cable Fantasy Shows) also get just as dark. But, they aren't non-stop bleak, they break up the dark with jokes and good times, so, although you may go on an emotional roller coaster, you have feel good moments too. Hell, even Once upon a Time, is much darker then it appears at first glance, and that's on ABC(Disney). So, I don't think dark is a problem, balancing that drama/sadness with hope and humor and good times, keeps them fun, despite the darkness. So, as long as they don't dwell in the pits of despair or crack inappropriate jokes at the wrong dramatic moments, darkness isn't a problem for me, and I think in today's TVscape, you do need some. Someone mentioned above the problem with screwing over likable characters over and over again, it's just as deadly not to have enjoyable characters for fans to latch onto (Enjoyable characters don't always have to be heroes, they can be villains [Dukhat, Kai Winn, Jeffrey Combs' characters])
It should be light in its optimism for the future and the great advancements humanity can make (not technological), but it shouldn't shy away from serious/dark storytelling. The characters should have backstories and flaws, so they aren't quite so squeeky clean as TNG. DS9 had a great feel to the series, but that was partly because of the setting and sets.
Space is disease and danger wrapped in darkness and silence. With a side of levity and hope. And lotsa mayonnaise.
Of course McCoy was talking figuratively, but he was referring mainly about the countless alien diseases that space travelers pick up from one world and then spread to countless others across the Galaxy. Imagine all the diseases on Earth and then multiply them by the number of inhabited known worlds and you get what he was on about..
Making it "dark" is to go against what Star Trek is. It can have "dark" elements, but that shouldn't be the sole focus, they should be kept in the background.
I think a certain amount of dark plot elements are necessary. TNG's biggest flaw is that they were in a sterile plot bubble where nothing bad could happen ever and everything was perfect and would turn out just fine the vast majority of the time. DS9 managed to find a great balance between being darker yet still maintaining the hope for a better future when the perfect society is conflicted with an imperfect world. Still, the show managed to have it's light hearted episodes. Voyager too found a different balance. While they retained the "things mostly turnout alright" element of TNG and TOS, they had some definite dark moments though they were often removed by the reset button. Things go wrong in real life and what makes stories interesting is how people deal with that. It's easy to be a "perfect human being" when everything goes perfectly after all.