How can I get my headset to work with my MacBook Air? (13-inch, Mid 2013)
You can use the earphone portion of the headset by plugging the appropriate cable into the audio output jack on the Air. The Air does not support plug-in microphones as it only has 1 analog output jack. The boom and Startech splitter will not be of use in this configuration.

My recommendation would be to purchase a Mac-compatible stereo headphone + mic to USB adapter. That would enable you to enjoy both quality audio and benefit from the close-field mic boom on the headset. Alternatively, you can just use the headphone part for listening and the built-in mics for speaking.