When I right click in the Solution of a C# Visual Studio project and select Add... > Class... it creates a class without a public modifier. How do I get Visual Studio (2008)
In addition of JochemKempe great answer, here is how to do it for the .NET core templates.
Instead of editing the templates inside the \CSharp\ directory, you need to edit the ones inside \AspNetCore\.
VS2019 (Enterprise): C:\Program Files (x86)\Microsoft Visual Studio\2019\Enterprise\Common7\IDE\ItemTemplates\AspNetCore\Code\1033\Class\Class.cs
VS2019 (Professional): C:\Program Files (x86)\Microsoft Visual Studio\2019\Professional\Common7\IDE\ItemTemplates\AspNetCore\Code\1033\Class\Class.cs