I'm going to limit my choices to actors and actresses who were working in tv in the 60s. So no Laurence Olivier as Chekov type casting.
Kirk: Robert Conrad for the action scenes and ladykiller stuff. Or maybe Robert Culp. Culp was a very underrated actor. I didn't appreciate how good he was until I saw him in a couple of episodes of The Outer Limits.
Spock: Donald Sutherland is actually a very interesting choice. I like that idea. My own pick would probably be Martin Landau.
McCoy: Joseph Cotten was doing a lot of tv work by this point in his career. Very talented actor and could pull off the southern accent. Or maybe William Conrad? A fat, irascible, but wise MD. Since he's afraid of the transporter he could have his own shuttlepod/'73 Lincoln Continental. No?
Scotty: Alan Young (He could do a great Scottish accent)
Uhura: This is a tough one. For some reason, I like the idea of the communications officer having language skills or at least a cool voice. I thought about a race or gender swap for this character. Ross Martin was a polyglot and had good chemistry with Robert Conrad. On the other hand, Brock Peters is a good actor with an interesting voice.
Sulu: Keye Luke (I like the idea of having a slightly older cast member)
Chekov: Lee Meriwether (I also like the idea of having very hot cast members. Think Miss Kitka from Batman '66)
Janice Rand: Barbara Eden (See above)