How to enable debug for site user in Salesforce

1. Set a browser cookie:

• Open your Force.com site URL in your browser tab. (This is important - You won't see debug log until Setting cookie without opening your site)

• Open the Chrome DevTools Console by pressing Ctrl+Shift+J (Cmd+Opt+J on macOS).

• Execute a command to set the cookie. (Copy and paste the below command and click Enter)

          If you use a .force.com domain, use this command.

         document.cookie="debug_logs=debug_logs;domain=.force.com";

         If you use a custom domain (for example, yourCustomDomain.com), use this command.

         document.cookie="debug_logs=debug_logs;domain=yourCustomDomain.com";

2. Find the name of your site’s guest user:

• From Setup, enter Sites in the Quick Find box, then select Sites.
• Select your site from the Site Label column.
• Select Public Access Settings | View Users.

3. Set a user-based trace flag on the guest user.

• From Setup, enter Debug Logs in the Quick Find box, then click Debug Logs.
• Click New.
• Set the traced entity type to User.
• Open the lookup for the Traced Entity Name field, and then find and select your guest user.
• Assign a debug level to your trace flag.
• Click Save.

Reference:

https://releasenotes.docs.salesforce.com/en-us/winter17/release-notes/rn_forcecom_debugging_guest_user.htm