Vysor puts a fully controllable window of your Android on your desktop
vysor.io
vysor.io
Regarding security, sometime back in Android 4.x the ADB protocol was updated to include public-key authentication. So it's (relatively) safe to keep debugging enabled now, even when you are plugging in to an untrusted USB port.
I think it's done in a similar way to SSH, i.e. the first time that a new computer wants to debug your device it supplies a public key and your Android device shows a popup with the key fingerprint and allows you to accept/reject the debugging connection.
edit: on a second though: what are the security implications? anyone done some research on this?