The short of it is Star Trek, created entirely from scratch today, would be completely unrecognizable. Too much of the show and what came after it was a result of the timing.
The semi-long of it:
Without a war with a draft, Gene Roddenberry has no reason to join the military. Subsequently, not being ex-military means he's unlikely to have wanted to become a cop. Gene Roddenberry still became interested in the emerging television market so a Gene who lived later would still be interested in television. There are two types of people who want to tell stories in film, those who have lived through stories they want to share and those who just grew up watching stories. Given all of Gene's problems and inhibitions, he'd be far more likely to be the latter in today's environment.
Also, content in television is freer now than it was in the 1950s and '60s, so he wouldn't have had to worry about masking the types of stories he wanted to tell as much as he did back then. There would be far fewer "message shows" if any at all.
Not to mention the space program was big in the '60s, as opposed to now, and Star Trek redefined space series on television. Without Star Trek, all you really have is Lost in Space and a lot of crappy sci-fi shows from the '70s and '80s that never lasted. And without TOS you don't get TNG which, to this day, I believe is responsible for making the flood of space-based series in the last two decades possible. Including the new version of BSG.
So you have a TV landscape where space sci-fi is lower regarded and even more niche than it is now. More than likely Star Trek made today would be a crappy show, with no message, that would air on Sy-Fy.
Gene Roddenberry's original idea for Star Trek was "The Cage", so we'll use those characters as a starting point.
Pike is cut from the same mold as Kirk except more serious. He could stay. In the 1960s, Spock inherited Number One's intelligence and cold demeanor. In 2010, Number One would stay Number One, so Spock would just be the alien who smiles and screams "The Women!!!"
Without the actors', the interaction between McCoy, who would still be Boyce, and Spock would never develop... not that it would even be the same Spock anyway.
The Klingons and Romulans, who were analogues from the Soviets and Chinese, would be obsolete.
With story arcs, the Enterprise might discover a "strange new world" every half-season with the crew spending several episodes in a row around a certain region.
And, of course, there wouldn't even be a guarantee that the Enterprise would even be part of Starfleet, a.k.a. the navy in space. It might be a different type of ship on different types of missions because who's to say a military ship would be the first thing to spring to mind for someone who wasn't in the service? For all we know, it could be a crew of space-truckers like on Alien,or just a group of scientists in space.
There are several factors for why I believe a Star Trek created from scratch today would be completely different.