At this time, the external case form in Customer Center cannot be set up to allow customers attach a file when submitting a new case.
Enhancement 118377 has been logged for this request. As an alternate solution, users can create an online case form and set this up as the default case form in the Customer Center.
1. Navigate to Setup > Support > Online Case forms > New.
2. Choose Default Form Template.
3. Provide a Title.
4. Mark the Enable Online checkbox.
5. Under Select Fields tab, insert the File field.
6. Insert additional fields as required and drag/drop the inserted fields to get the desired layout. For additional information about setting up an online case form, see Online Case Forms.
7. Click Save.
8. Navigate to Setup > Support > Support Preferences.
9. Under External tab, set the newly-created form as the Default Customer Center Case Form and click Save.
Customers will now use the online case form in Customer Center and have the ability to attach a file when creating new cases.