The pop culture understanding of Captain Kirk is that he was hopping from planet to to planet, doin' it with any green-skinned alien he could find. But about 2/3rds of the way through the first season, the truth is subtler and more interesting -- he's a character of barely contained sexual desire. Early on he confides in Dr. McCoy (not yet rebooted as Karl Urban, still the pleasantly, crotchety DeForest Kelley) about his secret yearning for Yeoman Rand, the lady with the great hair who brings him his meals of brightly colored cubed food. Yeoman Rand clearly has the hots for Kirk, and when Kirk is split into two beings in "The Enemy Within" his "bad self" tries to force himself on Rand. But when he's a complete being, Kirk is committed to his ship -- even though he clearly wants more, kind of from everyone, kind of all the time. The best example is the infamous back rub scene from the cold open of "Shore Leave"...
...in which our Captain enjoys what he thinks is a vigorous backrub from Mr. Spock, then immediately loses interest when he finds out its from unnamed Yeoman. No wonder there's so much slash fiction on Spock and Kirk. They're pretty smoldering together.
It's not just Kirk. One of the earliest episodes to air was "The Naked Time," which saw the crew with their most secret desires unleashed. It's the episode where the iconically greasy and sword-wielding Sulu comes from, but even before that we see Sulu trying to sell another male crewman on the benefits of fencing as they tend to alien plants -- one of which is clearly someone's puppet-hand. There's just a tension in the air on that ship, man.
Those are some of the instances when the sexuality of the Enterprise is overt. In every episode you get soft-lighting, eyeshadow on everyone, big leather boots, dark pantyhose and little skirts. In "Arena," when McCoy announces that he's looking forward to a meal that's cooked and not "reconstituted," Spock accuses him of being a sensualist -- to which McCoy responds "You bet your pointed ears I am."
There's no doubt that a big source of the under-the-surface sexuality from the original Star Trek is Spock -- and I haven't even gotten to the second season's "Amok Time" when Spock goes into heat. What the most recent Trek movie does to turn up the sex is to marry Spock's devotion to logic with Shatner's version of Kirk and his constant struggle to keep is sexual desire in check. I think it makes them both less interesting characters.
I had a passing familiarity with the original Star Trek as a kid. My brother was a big fan, but I only became invested in it with The Next Generation series. I would sit in a lounge chair with a pillow across the arms and pretend to pilot the ship -- just like Data! -- every time it was on. I would get so excited to be watching TNG that I would literally run circles around the room during the commercial breaks, to the point where my mom would threaten to turn off the TV if I didn't calm down. But it was pure nerdy excitement -- as much as I love TNG to this day, I can't say the show was all that sexy. Even if I had a big crush on Lt. Tasha Yar, those uniforms just don't have the same base appeal as the original.
And even in the episode where Riker falls in love with one of the genderless J'naii, it's nowhere near as kinkily titillating as Kirk hiding in a cave from the dude who played Lurch, waiting to attack him with ... um, a stalagmite.
It's impossible not to lean forward and beg these impossibly restrained, impossibly attractive, Starfleet officers to just get it over with and kiss already. Each other, aliens, anyone. Though it's true that Kirk does give in to those baser desires from time to time -- including the famous, first-televised interracial kiss between Kirk and Uhura in the third season episode Plato's Stepchildren -- it's the restraint and the innuendo that makes the original series so enjoyably, tensely sexy.
When the 2009 reboot put Kirk right into bed with a green alien, it was direct and it was used for comedic effect. It played on our memories and feelings about Star Trek more than it respected or recreated what Star Trek really was.
And worst of all? It just wasn't that sexy.