It's not much different for classes. The body of the constructor function simply becomes the body of constructor:
class MyClass {
constructor() {
var privateFunction = function () {
return 0;
};
this.publicFunction = function () {
return 1;
};
}
}
Of course publicFunction could also be a real method like in your example, if it doesn't need access to privateFunction.
I'm not particularily advising to do this (I'm against pseudo privat properties for various reasons), but that would be the most straightforward translation of your code.