I have not tried this myself, but there's an app named ADB Wireless (no root) which, as the name suggests, claims to make this possible without involving root:
This app is for developers wanting adb access over WiFi without rooting your phone.
*** NO ROOT REQUIRED ***
Usage instructions can be found on the app's playstore page – looks pretty easy and straight. And the permissions look phantastic: nothing that's not necessary or explainable. In fact, just a single one to view WiFi connections. So no ads or spyware to be expected ;)
Same to be said for an app called ADB Wireless (no-root) (yes, same name – different dev), which seems no longer maintained though (last release was 6/2013):
the first android application in the Google Play store to provide a wireless ADB connection without root.
Asks a few more permissions (a.o. full network), though.
Note however that all those non-root solutions seem to require a (USB) cable connection for initial setup – so there's probably nothing for your special case.
No app for that if your USB port is busted. So unless you get your busted port repaired or root your device, I see no way to get ADB running wirelessly. As initialization requires more permission as any "standard user" on Android gets, there are no command-line tricks either unless you already have root access: the initialization via USB makes use of the ADB daemons elevated permissions, which are only accessible that way.
For other readers:
Initial phrasing of my answer omitted that fact, as the OP's USB port is broken/unusable: If your USB port is in working order, and you can use an USB cable to connect to it, you won't need any of those fancy enabler apps (see here for more) just to be able to use ADB wirelessly (though some might come in handy if they e.g. offer a tasker plugin, so you could enable wireless ADB in your home network automatically and turn it off when you leave it). As Firelord pointed out in his commend, and Lucky linked to a post on SO, steps to enable wireless ADB are easy then:
- enable usb-debugging on your device
- connect your device to your computer via USB cable
adb tcpip <IP address>:<PORT> (e.g.
adb tcpip 192.168.1.50:5555 – find the correct IP address in the wireless properties of your Android device) on your computer
- still on your computer, run
adb connect <PORT> (e.g.
adb connect 5555)
- unplug your USB cable, it no longer needs to be attached – ADB now runs via WiFi