Photoshop Elements 13 [on Windows 10] shutting down after cropping a photo.

Try this:

Restore default preferences
Preference settings control how Photoshop Elements Editor displays images, cursors, and transparencies, saves files, uses plug‑ins and scratch disks, and so on. If the application exhibits unexpected behavior, the preferences file could be damaged. You can restore all preferences to their defaults.

Note: Deleting the preferences file is an action that cannot be undone.

Do one of the following:

Press and hold Alt+Control+Shift (Mac: Option+Command+Shift) immediately after Photoshop Elements begins launching. Click Yes to delete the Adobe Photoshop Elements settings file.
Go to Edit > Preferences (Mac: Photoshop Elements > Preferences > General), click the Reset Preferences on next launch button, and then click OK. When you restart Adobe Photoshop Elements, all preferences are reset to default settings.
A new preferences file is created the next time you start Photoshop Elements Editor. For information on a specific preference option, search for the preference name in Help.

from here:

https://helpx.adobe.com/photoshop-elements/kb/preference-file-locations-photoshop-elements.html

Does it do that every time you try to crop?

Windows 10

Photoshop Elements, Version 13.0.

Presumably you mean Photoshop Elements? Photoshop is a different app.

Which version of Photoshop Elements? Which operating system?

Any error messages or crash reports?