Yes, the latest version of Yamaha's USB-MIDI driver works with the PSR-E403 and is available for either 32-bit Windows 10 or 64-bit Windows 10-- just make sure you download the right one for your computer (i.e., 32-bit or 64-bit). Do not even try to install the driver that came on a CD-ROM with the PSR-E403, because it is a really old version that won't work under Windows 10 (assuming you can even get the installation program to run).
Note that in order for the driver to recognize and work with any Yamaha keyboards, you must have the keyboards connected to the computer and turned on before you install the driver. That means if the driver was already installed on the computer for a different model of Yamaha, you may need to uninstall it (which is done through the Windows Control Panel), then connect all keyboards and turn them on, then reinstall the driver so it sees all the keyboards it needs to work with. The only exception is if you have models which use the same description as each other-- e.g., the PSR-E403 is described as a "Yamaha PortaTone" whereas the PSR-E433 through the PSR-E463 are described as a "Digital Keyboard." (The description I'm referring to is shown enclosed in square brackets at the top of the MIDI Implementation Chart in the keyboard's manual.)
Anyway, I have a YPT-400, which is another name for the PSR-E403, and I use it with 64-bit Windows 10, so I'll be happy to help your cousin if there is a problem.