With CGI, they can take Kirk's face (they have oodles of footage), and place it on another actor post production. I'm sure they could do the voice too. In fact, not just Kirk, but all of them. It would be rather expensive and time consuming though, but ok for a small screen television show.
Deep Space Nine used old footage from the Trouble with Tribbles, so a precedent has already been set. It would be criticised as a blatant attempt to win back popularity for the show, especially by harsh critics, so if it wasnt done really well to be 'worthy' of the historical cultural icon that its become, it would probably backfire and even the fans would be discgusted. (Look at the reaction to Enterprise).
The Original series is still my favourite, despite its archaic special effects and 60s styling.