I have a Web API (all my code) method that I want to be called on a schedule from within Azure. I have it all working. I can specify the URL, set up a schedule and it works fine. What I want to do though is restrict the call to a certain 'system' user.
All other Web API methods are called from a Web Site. The Web site allows the user to login and receive an 'access token' and then that token is sent along with all other requests. So a two-step process. This all works fine too.
How do I pass the 'system' user/password to the Web API method from within the Azure scheduler? It looked pretty straight forward, choose Basic Authentication and then enter the user/password combination. It still calls the Web API method, but its not authenticated? I'm not sure how I get the User authenticated 'before' calling the Web API method?

You probably use an Identity Provider like Azure Active Directory. You should use a service principal which represents your schedule application that is allowed to call your API instead of user principal (your system user).
Read more: Application and service principal objects in Azure Active Directory (Azure AD)
So within the Authentication Settings you should choose Active Directory OAuth and provide the neccessary values:
Basic Authentication must be configured within your WebAPI, its not related to the token authentication which you are using.