I prefer production order, especially in the first season where it just makes more sense (e.g. Kirk's complaint about Rand in "The Corbomite Maneuver" makes more sense if she's new than if they've already had a bunch of episodes together), and where you can see the real-world process of the writers making up the universe as they go along. I've never seen a reason to favor airdate order, because the airing order wasn't based on any story considerations and was fairly arbitrary -- it was a mix of which episodes the network wanted to lead with (e.g. season 1 started with an Outer Limits-style monster episode because that's what the network expected science fiction to be, and seasons 2 & 3 started with Spock-centric episodes because he was the breakout star) and which episodes took more or less time to complete the effects for. It may not make much difference, but there's nothing about airdate order that materially improves things; it's just a pointless appeal to nostalgia, which makes no sense because the majority of viewers alive today never saw it in first run anyway.
It's weird how the preference has shifted, though. When I was a kid, the syndicated episodes seemed to be run in arbitrary order; at least; I remember seeing a Chekov episode only a little while after seeing my first episode, "The Corbomite Maneuver" (and my 5-year-old self was scandalized that they'd give a character an unreal, ridiculous name like "Mister Check-off"). The Blish adaptations were in seemingly random order. The first major reference books -- The Making of Star Trek, The World of Star Trek, and the Star Trek Concordance -- all listed the episodes in broadcast order. But then in 1979, Allan Asherman's The Star Trek Compendium listed them in production order for the first time, and listed "The Cage" as a separate episode from "The Menagerie" for the first time (the pilot had actually been renamed "The Menagerie" by the time it was filmed, so "The Cage" was only unofficially used to differentiate it until it was codified by the home video release). The syndication package also got re-released in production order at some point, and for the next couple of decades, every reference text and home video release used production order as well, so it became the accepted default order. Even the re-release of the Blish adaptations in a three-volume set, one thick paperback for each season, reprinted them in production order (aside from Blish's "The Menagerie" actually just adapting "The Cage"). So it was weird when the Blu-Ray release suddenly reverted to broadcast order and lots of people seemed to jump on the bandwagon for that. Where had this desire for airdate order come from after a whole generation had grown up knowing only production order?