Ninject and static classes - how to?

Don't do it. Don't use a static class that needs dependencies of its own. This makes testing harder and other types that depend on this AuthenticationHelper won't be able to include it in their constructor which means they hide the fact that they depend on it.

Instead just do what you would always do: make AuthenticationHelper non-static, implement an IAuthenticationHelper interface on it and inject all dependencies through its public constructor.

But if you insist into keeping that class static (which again is a really bad idea), create a static Initialize(UserBusiness userBusiness) method on it, and call this method in the start-up path of your application. You can't let your DI container call this static method. They don't allow because 1. it's a bad idea, and 2. such static method only has to be called once, so letting your container auto-wire this for you doesn't really help.