1) I'd make it a total reboot of the franchise. Start with a TOS-equivalent, but clear out the entire timeline so that it can do whatever it wants.
2) No world war III, no Eugenics wars, no century of horrors. They're unlikely to happen in real life, no need to have them in the timeline. In fact as a general rule I'd tell the writers to avoid giving the viewer information about things that happened on Earth until after the year 2100. No "When Mars was colonised in 2020" lines which are just going to date your show.
3) I'd keep TOS where it is, in the 2260s. But I'd make the Federation and space exploration in general younger within the timeline. The Federation would be a looser alliance, far more akin to the EU than the USA, an alliance of sovereign states. Maybe even looser than that, something closer to the UN now. The Federation would pre-date TOS by no more than thirty years, and interstellar travel by no more than 70. I would want people alive who remembered the first Human warp drive, and most adults would remember a time before the Federation existed.
4) No universal translator, as others have said. BUT... there would be an common language. Let's call it "Inter Stellar Basic." ISB is the "Lingua Franca" of space; most species capable of interstellar travel know of it and most ship crew on interstellar ships speak it. You can buy translators that convert it to english for a few spacedollars. The point of this is that on the one hand, you can depict the reality that meeting a genuinely new species brings language difficulties with it. But on the other, it's not so simple as everyone speaking English - though as a convention, ISB may be rendered as English on screen just to avoid all those tedious subtitles or an interpreter endlessly translating lines.
5) No prime directive. At MOST, the prime directive would be a Federation policy that Starfleet may not interfere in the politics of other worlds without specific orders. But leaving people to die rather than help - no. No way. And yes, the Federation would intervene when it felt the case was right. The reason being that we've done the "we can't interfere!" thing to death, let's do something new!
6) Generally a more realistic mixture of technologies. By which I mean, computers way more advanced than Trek typically shows them. Plus lots of genetic engineering, highly advanced medical science, etc. But interstellar travel is relatively new, it's difficult, it's hellishly expensive. I want the Enterprise to feel like it's on it's own out there most of the time, in a dark, scary place where help is not usually available. Think about travel during the age of sail, where journeys took weeks, months or even years. No more nipping home for the weekend!
And no replicators. You can have machines that make things, and they can make things quickly and easily, but they are more like distant descendants of 3D Printers than they are like replicators. They're seriously limited in what the can produce.
7) Nothing about Humans "evolving" beyond basic emotions like greed. Human nature hasn't changed in thousands of years, it won't change in the next few hundred. In relation to this, the Federation is NOT a post-scarcity society. There is money, people are paid for their work, the Enterprise captain has to be aware of his operating budget.
8) I'd like to see a more realistic depiction of weapons and their firepower. TOS did this better than anything that came afterwards. Being hit by a ships' phaser bank or a photon torpedo should be like being hit by a nuclear blast. A LARGE nuclear blast. If the shields go down and the ship gets hit again, the ship is vapourised. Totally. And weapon ranges are in the hundreds of thousands of miles - two ships next to one another exchanging fire is something we should never see.
Similarly, decide how powerful your hand weapons are and stick with it. Don't treat a hand phaser like a handheld artillery piece in one scene and a .45 in another.
9) Echo what people have said about more diversity. Let's face it, at least 85% of the cast should be from places like India and China. And yes, some of them should be gay, some of them should be transgendered, etc.
And NONE of this should have a "story of the week" about it! Do NOT give us a story with the moral "it's okay to be gay" - simply show us a society in which nobody thinks anything of it if you're gay.
That's all I can think of for the moment.