MacBook Air Wi-Fi is not working -- how can I troubleshoot the problem?
It sounds like that your network settings are corrupt. 

Reset your wifi settings
Another option could be to reset the network and wifi settings. 

a) In Macintosh HD/ Library / Preferences / SystemConfiguration 

com.apple.airport.preferences.plist


 


Move those files to the trash
Reboot your Mac
Reconnect to your wifi


b) Alternatively you can open SystemPreferences > Networks > select Wifi and delete it. In the second step re-add the wifi service again.

Upgrade to macOS
If possible try updating your Mac to the latest macOS version. Open the AppStore and download Mac Sierra.

Disable Third-Party Software
Do you use any kind of Firewall Apps e.g. Little Snitch. If so, temporarily disable them.