I currently have all my common utility function methods in a base controller, which all of my controllers inherits from. These are methods for functionality like uploading files, resizing pictures, deleting files, sending e-mails, generating random passwords, hashing passwords, etc.
What is the recommended structure for these kinds of things?
In that case you shouldn't put all these utility functions inside your base controller. You will face a lot of problems if your project grows. Changes and testing of these methods can become difficult, all your inherited classes need to use the same utility methods, etc. Have a look at Composition over inheritance to see another approach.
Somethimes I tend to organize my projects in the following manner if that might help you:
Simple helper methods: Create a folder and a namespace (e.g. namespace [...].Common") inside your web project and put one or more public static classes inside it (e.g. class "FileHelper.cs", "StringHelper.cs", etc). If you need one of these methods in a controller action simply put a "using ...Common" statement at the top of your controller class and call e.g. FileHelper.MethodName.
If I can define a closed subject area with a group of methods I try to encapsulate these methods in a service class (maybe even outside the web project if I have a feeling that I might need this functionality in other projects too), define an interface for that class and plug that functionality into controller classes by using dependency injection. If you don't know about that concept you should definitely read Dependency injection in ASP.NET Core. Dependency injection is a vastly used major concept in ASP.NET Core projects that brings you a lot of advantages and - if used correctly - stears your work into well-organized projects.
More complex organizations are always possible depending on your needs. Have a look at multitier, hexagonal or onion architecture if your projects grow.