Without the original series, Star Trek would not exist. That isn't much of a stretch to say. Nonetheless, over the past 59 years, that series has ultimately spawned a massive franchise with a vast, fertile tapestry of interesting stories featuring compelling characters and exciting settings. This has led me to wonder: What if things were different? What if we mixed up what we know? Let's speculate.
This first imagined mash-up we will consider is what it might have been like if James T. Kirk had commanded Deep Space 9. In the eponymous series, DS9 was commanded by Commander (later Captain) Benjamin Sisko and became embroiled in politics, religion, and war as it sat at a critical nexus between Bajor, Cardassian Space, and a wormhole to the Gamma Quadrant (and thus the Dominion).
Captain Kirk, of course, is the iconic captain of the USS Enterprise (NCC-1701), originally portrayed by William Shatner and performed in more recent years by Chris Pine and Paul Wesley. For the purposes of this speculation, we will focus on Shatner's version of Kirk, as there is considerably more to go off of, and his performance is the quintessential Kirk.
How would Kirk affect DS9?
As a space station, most of the adventures found on Deep Space 9 come to it. This means most encounters will be with people from warp-capable societies. It also means that diplomacy plays a much larger role than science or discovery. As an adept diplomat—even if he wouldn't admit it—Kirk would likely do that job well, but maybe in a different way from how Sisko did it in canon.
One thing that Kirk would likely do differently is that he would probably be more assertive towards the Bajoran government and religious figures, like Kai Winn. Of course, Sisko's status of Emissary legitimized his presence in the eyes of the Bajorans, and Kirk would not have that in his favor. In some ways, though, that label was as much trouble as it was help for Sisko.
Kirk is quite adept at flexibly understanding and respecting the beliefs and practices of other cultures while also speaking up when something is harmful or exploitative. I could imagine Kirk standing up more directly against a figure like Kai Winn, despite her status on Bajor, and highlighting how Winn was manipulating and exploiting Bajorans for her own benefit.
Of course, this could either do away with the troubles that Winn causes people on DS9, or it could lead to more trouble. That would really depend on how receptive Bajorans would be to a non-Bajoran (with no special Emissary status) questioning their religious leader. And note that they may not be very receptive at all, given their very recent history of occupation by the Cardassians.
That said, Kirk's ability to understand and empathize with others means that his fighting tactics are different. There is often less fighting and more finding unique ways to win without fighting. In the Dominion War, this could mean Kirk finding ways for the Dominion to hoist themselves by their own petards.
How would DS9 affect Kirk?
It is well established that Kirk's place is as captain of a starship. A space station is decidedly not a starship. Although Deep Space 9 certainly encounters its fair share of strange phenomena and new cultures from the Gamma Quadrant, it seems unlikely that Kirk would have been content to simply wait for things to come to him.
That being said, commanding a station like DS9 may still be more palettable to Kirk than the job he had as an Admiral at the head of Starfleet Operations in the early original-cast films. Rather than planning fleet deployments, Kirk on DS9 would be flexing his diplomatic muscles, especially as he would navigate Federation relations with Bajor, and potentially averting the conflict that was the Dominion War.
At the end of the day, though, command of DS9 is more of an administrative position than regularly warping to new planets and beaming down to discover what's happening. Kirk's proactive nature to seek things out would probably make commanding DS9 difficult for him, even if he is good at the actual job.