I have a ASP.NET MVC (NOT ASP.NET Core) single page application with angular js on the front end.
My client (browser) talks to server
First of all when you remove login and there's no authentication mechanism in your application, there's really no way to secure anything, because anyone can access your APIs. I think what you want is to make sure that your APIs are called only from your own website. Unfortunately you can't completely achieve that, since your web APIs are http/https, and anyone, from anywhere (like postman, fiddler, ...) can create a http request and call your API.
All you can do is to make it harder for your API to response to requests, like using Anti-Forgery as you mentioned.
And also I suggest you add a cookie for your application and check that cookie in every request, in this case it's more complicated ( not impossible ) to call your API using Fiddler or Postman.
And last I suggest that you use CORS, so browsers would only allow your domain to call your APIs. So nobody can call your APIs in a browser from different domain.