First we need a program to map controller outputs to keyboard keys. An example of that would be joy2key or what i use joycur (http://www.deinmeister.de/jct_e.htm)

In joycur (or similar) you want to add WASD twice and check the checkboxes for the correct positions for your stick and your dpad. Specifically in joycur you want to disable "use typematic". If joycur tells you that a joystick key or axis is not released, tell it to exit anyways. In PSXJIN you want to set your movement to WASD and you should be good to go.

Windows 10 only: To fix the issue of windows 10 registering the trigger buttons as always pressed you want to add both of them to keyboard buttons, it should be the opposite of what is shown as pressed at the very top (T and L should be pressed by default which is the problem, B and H should be what your keys are mapped to). Now the tricky part is to get PSXJIN to accept that, either you unplug the controller and set everything up via joycur including all other buttons or you can try to press both trigger buttons about half way in so it will register them as neutral and not overwrite everything you click.

Bizhawk ignores this issue so this workaround is not needed.