How to pass parameters to anonymous class?

Technically, no, because anonymous classes can't have constructors.

However, classes can reference variables from containing scopes. For an anonymous class these can be instance variables from the containing class(es) or local variables that are marked final.

edit: As Peter pointed out, you can also pass parameters to the constructor of the superclass of the anonymous class.

Yes, by adding an initializer method that returns 'this', and immediately calling that method:

int myVariable = 1;

myButton.addActionListener(new ActionListener() {
    private int anonVar;
    public void actionPerformed(ActionEvent e) {
        // How would one access myVariable here?
        // It's now here:
        System.out.println("Initialized with value: " + anonVar);
    }
    private ActionListener init(int var){
        anonVar = var;
        return this;
    }
}.init(myVariable)  );

No 'final' declaration needed.