I really enjoyed the old show, but I don't think that too many influences would change... I guess there'd be a lot more "planet of the apes"-style tattoos, which could be a good thing.
I will abstain from the "monkey on your back" puns that come to mind.
My teen years would probably have been a little different - referring to something else rather than next gen... maybe the new twilight zone?
I like my current world, since I know it, but would be happy to voyage where no man/one has gone before.... erm, journey into the unknown for a little while.
I'm not sure what would replace Checkov's "kyptin/ARGHHH!", Kirk's KHANNN!, and Dammit jim, I'm a doctor, not a ... and the term "redshirt".