During the adventures depicted on screen, we only got evidence of three counts of sexual intercourse: with Carol Marcus to give David as per ST2:TWoK, with Miramanee in "Paradise Syndrome" (but the mother and child died), and with Deela at gunpoint in "Wink of an Eye" (with every indication that, if the biologies were compatible to begin with, the child would be carried to term). That is, during TOS, Kirk never voluntarily engaged in sex, as far as we can tell.
In "Bread and Circuses", Kirk's enemies offered him the services of a slavegirl, but Kirk would have had every incentive to refuse. And in "Elaan of Troyius", he managed to resist the temptation of Elasian pheromone tears. He was generally a very private person, readily using his charms to manipulate women in order to complete his Starfleet mission or to protect his ship but seldom engaging in relationships otherwise. The jury is still drinking over the issue of whether "Requiem for Metusaleah" falls in the category of seduction-to-save-the-ship or represents a rare, indeed almost unique case of the Captain actually having a crush.
The older Kirk of the movies apparently had relationships more freely than the TOS era hero: we saw Gillian from ST4, and heard of Antonia at some point in the past. No sex with the former, tho, and one would think the issue of children would have come up when Kirk relived his days (years?) with Antonia in ST:GEN if that liaison indeed resulted in any.
Timo Saloniemi
