The composer is likely Gerald Fried, but could have also been Sol Kaplan.
Apparently, the majority of the music in this episode was recycled from other episodes. This was done frequently to save cost and time. This cue in particular could have been heard in longer form in one of the episodes listed in the quote below or an unused portion from one of the scores (would have to go back and listen to each one to be certain)
Gerald Fried got the music credit. Most of the cues came from his
scores for "Catspaw", "Amok Time", and "Friday's Child". A bit of Sol
Kaplan's "Doomsday Machine" music was thrown in for good measure.
There was one piece of new music from Fried. As Kirk tell Scott to
"beam us home", the music is somber and poignant. It is also unique to
the series and is not heard in any other episode.
source: These Are the Voyages - TOS: Season Two - Mark Cushman
For what it is worth, the only reference to the word 'clarinet' in this book is found in the chapter for "Catspaw" which, as shown above, was composed by Fried.