The Enemy Within
Planet Surface
Known: Eddie Paskey
Unknown: 2 crewmen in sciences jumpsuits (#6, 62)
Based on the outfit, it's possible Eddie Paskey is playing Connors from "Mudd's Women" here. The crewman with thin brown hair appeared in "Where No Man Has Gone Before".
Bridge
Unknown: Communications officer (#63), helmsman (#64), crewman in operations uniform (#65), James (#29)
The red shirted crewman whom Kirk addresses as "James" could be seen in "The Corbomite Maneuver".
Sickbay
Unknown: Bobby
Bobby gets to autopsy the space dog.
Transporter Room
Known: Eddie Paskey (twice!)
Unknown: 2 crewmen in operations jumpsuits (#66, 67)
Corridor
Known: Ron Veto
Unknown: Helmsman (#64)
The crewman walking by in two scenes looks like Ron Veto, but I'm not absolutely certain.
After Fisher interrupts Kirk and Rand in her quarters, as Kirk is hitting him, someone can be seen running away in the background. It's possible this is Ed Madden getting out of the scene.
Kirk's Double
Don Eitner doubles for William Shatner in scenes that requires both Kirks to be seen. He will later appear in "Charlie X".