Jonathan Mori
Is that web tier authentication?
If so, you must have your server(s) federated with a portal. See the help link below. With web tier authentication, there is a limitation, at the web architecture level, that does not allow the user identity to flow through. As you know when you submit a request to a print service, like your web maps, it tries to access map/feature/image services to get image/features etc. In order to access those services, a print service must have privilege to access them -- since end users identity does not flow through, it basically relies on token. When your servers are federated to a portal, token gets passed around and print service can access secured services.
Print maps that contain secured services—Documentation (10.7) | ArcGIS Enterprise
Having said that if you don't have portal and your security requirement is not really a per user level security you can choose one of the following options as last resorts (neither of them are recommended approach and we ask you to opt in to this with ample caution).