(A) what main aspect/s of the 60's show would you change/update? — 180 Proof
Minute 1 of day 1, get rid of those piles of klunky hardware they substituted for computers. Even STNG still has half a dozen big tablet things to contain the amount of information a 300-year-old cellphone wouldn't even notice.
I'd certainly put the female characters in the same uniforms as the men wear and either give them all similar, practical haircuts or else show more versatility in all the crew's personal appearance - I lean toward the former. I'd like to see coherent, ready-for-action crew.
Seatbelts and lanyards. It's indecent how those poor people are made to bounce around the cabin every time they hit a space-pocket or enemy shell, and lose their weapons, tricorders or essential weapon at the first clumsy move. (Actually, I often wonder why so many characters in all kinds of drama keep dropping their phones down sewer grates, when all you need is a cord like boyscouts understood in 1908)
And for heaven's sake, I'd drop the attitude of "here's a planet we know nothing about. Let's just beam down there with no protective gear!" Space suits mandatory for initial survey!
I'd want the aliens to be a lot less obviously human. There are excellent makeup artists and animators out there, waiting for a chance to be creative. I appreciate that actors come in a limited range of sizes and shapes, and that characters need to fit in the set, but even within those constraints, their bodies, apparel and accoutrements could be more varied.
I can accept a universal translator - else scripts would be painfully awkward - but it should take a few samples and several minutes to turn unknown alien speech into English vernacular.
I also quite liked the novel serialization aspect of DS9 and Babylon 5. It's a good idea to have thematic lines and chronology and character development from which good writers can make engaging sub-plots for each episode.
(B) what you would keep from the original to retain its identifiably Star Trek style rather than feeling like another generic space opera cashing-in on the franchise brand with all that glittering s/fx, pointless techobabble & Mos Eisley "aliens"? — 180 Proof
I'm quite happy with the non-monetarist economy of Earth, but would need some kind of standard trading medium with other cultures. (Voyager bartered, and that's acceptable, but they shouldn't have had to improvise.)
I'd keep Star Fleet and maybe even the Prime Directive - though I'd either have to make it more flexible (twenty-seven pages of exceptions and special circumstances) or have the officers agree to consider one another's reasons for breaking it before going all legalese on his ass.
I'd keep the Federation, of course.
I'd keep the generally relaxed an homey atmosphere of the interiors: If people are going to live aboard for five years, they should not have to look at blue-grey brushed steel surfaces.
And for sure I'd keep the time travel. Those were some of the most fun episodes of all four series.