Apple A1277 USB ethernet adapter not working on MacBook Air
For reasons best known to itself, Apple chose not to include updated drivers for the A1277 adapter when it released Mavericks. In the screenshot you provided, the 'Product ID: 0x1402' in reference to your adapter reveals it was manufactured by ASIX Electronics, and further indicates that ASIX designates it internally as a model AX88772C. (This information comes from the massive OUI database of model and vendor codes maintained by the IEEE.)

Happily enough, even though Apple no longer continues to support the product, ASIX does. Their current driver version (2.2.0) for the A1277 / AX88772C includes support not only for Mavericks, but for Yosemite as well.

You will find the link to download "Apple Mac OS X 10.5 to 10.10 Drivers Installer" as the first entry in the list of driver downloads on this page of the ASIX site:
http://www.asix.com.tw/download.php?sub=driverdetail&amp;PItemID=136