I don't think there was a female (human or not) that Kirk didn't shag. Anyway, I always got the impression that Kirk did have a relationship with Rand. Regarding fraternization, if I remember my teenage reading of "The Making Of Star Trek" and other books like it, Rodenberry thought that relationships would be pretty common during a "5 year mission" on a co-ed spaceship of 400+ individuals. Besides, they hadn't invented "sexual harassment" yet.....
Did you know that the number of each characters romantic partners was actually stipulated in the actor's contracts? I don't remember the actual numbers, but it was some ratio like for every 10 episodes, Kirk got 5, McCoy got 3 and Spock got 1. Scotty, Chekov & Sulu got the leftovers. (and Uhura? Did she get any romance? Don't remember.)