If anything, this new Star Trek takes itself more seriously than did the campy Star Trek before it.
The more serious Star Trek is the sillier it becomes.
Y'know what kills me in Trek? When a silly, lighthearted episode has a conflict that might destroy the entire ship/station.
It was done well only once, with the bomb in Trouble with Tribbles. After that, it seemed that in order to have a few laughs meant that lives had to be in mortal danger.