There's an app called Remote Web Desktop by SmartDog Studio. It enables access to your phone through your desktop's browser, over the local network (i.e. no connection to the Internet is necessary (I use it with both phone and desktop on the same WiFi AP); phone acts as HTTP(S) server on non-standard port (configurable)).
The simplest method is "shared clipboard": select the text on your phone, copy to clipboard, and it's immediately accessible from the desktop computer.
The app has a free (ad-supported) and paid (~$2~$4 USD) version; the free version allows connection over HTTP, paid also offers HTTPS.
Full disclosure: I'm a (happy) user of this app, no other connection.