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".