Captain: James T. Kirk
Kirk is always my choice for Captain. While I like all three others, neither Sisko or Janeway are anywhere near Kirk and Picard, and of those two I feel Kirk has more consistent showings and lacks Picard's arrogance.
First Officer: Spock
Going with Spock to pair with Kirk is a no brainer. They are one of the best tag teams in Trek and iconic for a reason. Of the other options for XO, I actually feel Kira is the next best, as she brings a unique set of skills and perspectives, but she is still well below Spock.
Chief Engineer: Montgomery Scott
No two ways about it, he was by FAR the best Cheng in a Star Trek show. LaForge learned from him, and my runner up choice, Miles O'brien, is, let's be honest, is just Scotty as an Irishman.
Science Officer: Jadzia Dax
Hey, Spock doesn't need to fill both roles, even though he did on the NCC-1701. Jadzia brings a wealth of experience and her own unique flair to things in a way that neither Data nor Spock does.
Tactical Officer: Tuvok
Likely a bit of a surprise pick here, but Tuvok brings considerable experience to bear as well as the laundry list of Vulcan advantages. The other options are Chekov and Worf mainly, and while fun characters, neither one quite has the consistent track record Tuvok does.
Operations / Communications Officer: Data
Yes, I merged these two categories, that's for a simple reason: Comm Officer was only a duty station in TOS. In later series these duties were handled by the Ops officer. What this means is that without merging them your only choice for Comms is Uhura, and for Ops your choices are Harry Kim or Data. These aren't choices AT ALL, as everyone will then pick Uhura and Data because... why would you pick Kim? Anyway, while Uhura is a highly competent officer, I think Data brings much more to the table, making him the best choice.
Security Officer: Worf
This one has Chekov, Worf, and Odo really vying for it (Tuvok too if he's not chosen for tactical), and of those three, while Odo brings a LOT to the table, I feel his style is wrong for a starship. Chekov, while competent, doesn't bring much that's not already covered. Meanwhile Worf brings a good set of skills and sufficient paranoia to ensure the safety of the crew.
Doctor: The EMH
Bashir is to wet behind the ears in many respects, and while superhuman, he also simply lacks the breadth of experience of the other doctors. Crusher has a lot of experience but well... she never struck me as outstanding. McCoy seems to be the obvious choice then, especially given I'm already putting Kirk and Spock up, but the EMH from Voyager brings everything McCoy had and more to the table. So I feel he's the best pick.
Helm/Conn: Tom Paris
There's not as many dedicated helm officers to choose from. You have Sulu, Paris, and Nog. And let's be honest, Nog is a good kid, but pales in comparison to Sulu and Paris. So the choice is really between Sulu and Paris and of those two, Paris has shown a lot more flexibility and impressive tricks up his sleeve. It's a close choice though.
Morale Officer/Counselor/Bartender: Dr. Leonard McCoy
The issue here is you need someone who's good at all three. Ezri is a trained Counselor, but not exactly the life of the party, and Quark is a good bartender and kept his pulse on the morale of DS9, but a sympathetic ear and trained counselor he isn't. Guinan has the same issue Quark does, she's a good Morale Officer and Bartender, but she's NOT a trained Counselor. And Neelix is... well... NEELIX, he's of dubious skill as a morale officer, has absolutely no skill as a counselor and I'm not sure he knows Saurian Brandy from Romulan Ale. Old Bones though, he WAS trained as a Counselor, kept his pulse on his crew's morale time and again, and we ALL know he had FINE taste in liquor. He's a dark horse candidate that you wouldn't expect, but he fits the qualifications and does the job best.