Enterprise. I love the show, but...
Let me start with the characters (incidentally, I would leave the cast as is).
Captain Jonathan Archer, 42: Son of Henry and Sally Archer. The chip on his shoulder re: the Vulcans goes to their failure to alert Henry Archer that he was on the wrong track for development of the warp 5 engine, setting him -- and ambitions for space exploration -- back many years. And yes, he'll still have Porthos.
Commander Charles Tucker III, 31: XO and chief engineer. Born in Chicago, raised in Pensacola. He and Archer became friends while working on the warp five trials. Appoints himself unofficial morale officer.
Lt. Malcolm Reed, 35: Section 31 arranges his assignment as tactical officer aboard Enterprise, where his covert assignment is to gather information on the species they encounter. Archer will learn about his connections about six months into the mission. From then on, Malcolm's actions and motivations will be suspect. But as he comes to feel at home aboard Enterprise, he'll want to earn the crew's trust. Eventually, he will renounce S31.
Ensign Hoshi Sato, 25: Ship's linguist. Over time she will also take on the role of protocol officer. She has been trained in the martial arts and runs gambling operations when off-duty.
Sgt. Travis Mayweather, 29: MACO. Born on a cargo ship, he never saw Earth until he was 20 when he chose the military over life as a boomer. His father was disappointed that he wanted to leave, but understood his son's craving for something more. Mayweather's bland reaction to aliens and the wonders of space -- not to mention stories about species and how many planets he has set foot on -- will get under Archer's skin.
Doctor John Phlox, 49: Had a year of service with the Interspecies Medical Exchange, which helped expand his medical knowledge, but because the NX-01's mission is moved up, he misses out on a second year that would have expanded his experience considerably.
Envoy T'Pol: Ambassador Soval will persuade SFC to have T'Pol join the crew for the first year as a science advisor. Her presence rankles the crew, especially when they learn that her true role is to evaluate and report to Soval on the crew's activities (with an emphasis on failure). Over time she will become acclimated and realize that their missteps are no worse than those in the early years of Vulcan exploration.
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I'd keep the NX-01 crew complement at about 80 men and women. Crew members will get sick, maimed and die on this show. I would show how families keep in touch: video letters from home as well as real time visits across the light years.
The ship will return to Earth after nearly a year to restock supplies and replace injured and dead crew members and for debriefings. We'll see families and what civilian life is like.
I would be faithful to the "history" established in the "sequel" shows... No Romulan cloaking devices; (I might keep the Borg ep just because it is consistent with First Contact); no Ferengi, didn't like them anyway. No Klingons (was kind of sick of them by the end of DS9); I would have made more use of the Vulcans, Andorians, Tellarites and the Orions would have been the main bad guys. Might add some "new" aliens but they wouldn't have a significant role in events, which would explain why they aren't mentioned in "future Trek."