First JavaScript doesn't have Classes.

Second, your third option seems more rational to me, but it highly depends to your requirements as well. Also you should not be much worried about exposing the helper function. The pros of the solution totally justifies the compromise to me.

Third, your time as a developer is valuable; Don't make trivial tasks hard to implement, time consuming and more pron to human-errors. Simplicity of source code is a great feature itself.

You must add a function into your helper and use the variable of parent function.

var person_helper = function(person) {
    var parent = this;
    this.name = person.name ;
    this.isHisNameIs = function(name) {
        if(name == parent.name)
            console.log('OK');
        else
            console.log('NOP');                    
    }
}

http://jsfiddle.net/H4RsJ/6/

Callbacks are the functions that are passed as a function argument and are performed after a particular event such as resolving of a promise.

Helper functions are the normal functions that are called at any point of time when the code execution is taking place. Mostly, helper functions are wrapped inside another function.

Example of callback function:

const fun = (callback) => {
  setTimeout(callback, 3000);
};

fun(() => {
  console.log('callback function');
});

Example of helper function:

const factorialOfNNumbers = (...numbers) => {
    const helperFact = (n) => {
      if (n ===1 || n === 0)
        return n;

      return n * helperFact(n-1);
    };

  return numbers.map(n => helperFact(n));
};

console.log(factorialOfNNumbers(2, 3, 4));