The largest PSX emulator, ePSXe, supports analog controllers with the configuration options included with the software. Certain games may not work properly when you are using digital controls, so switching over to the analog control scheme is the best way to mitigate any issues relating to that. The configuration option takes a moment to set up. After configuration you can use the analog stick with no issues at all.
Click "Gamepad 1."
Click "Port 1," then click "Pad 1."
Select "Dual/Analog" from the drop down menu on this configuration page, and save your settings. Your analog controller will now be enabled in the software.