String Calculator kata in JavaScript using TDD

beforeEach(function() {
  calculator = new StringCalculator();
});

I usually recommend not using things like beforeEach (and similarly afterEach). The problem is that your tests are now implicitly dependent on an operation that's not referenced or even called anywhere in the test body. It's not immediately clear that a calculator was defined for each test. It's also not clear if it's a per test instance or a shared instance.

Tests should be independent and self-contained. The general rule I follow is I should be able to cut and paste a test to any part of the suite and still have it work, or remove it without affecting other tests. You can have shared utility functions that are called explicitly in the test body so that it's clear to the reader what happened, where something was obtained, etc.

function StringCalculator(){...}
StringCalculator.prototype.add = function(string_numbers) {...}

Although the rules mention "a method" which immediately would imply an instance and therefore classes, you don't necessarily need one. You'll only need a function that takes an input and gives an output. Keep it simple.

it("should return the sum of an unknown amount of numbers", function(){
  // creates a random array
  randomArray = (length, max) => [...new Array(length)]
  .map(() => Math.round(Math.random() * max));
  // creates a random number between 1 and 100
  randy = Math.floor((Math.random() * 100) + 1);
  res = randomArray(randy,randy);
  // sums up an array
  sum = res.reduce((pv, cv) => pv+cv, 0);
  // converts the array to a string
  arg = res.join();
  expect(calculator.add(arg)).toEqual(sum);
});

Never forget to use var, let or const if you don't want these variables to be defined in the global space (or at least the top level of the module). I usually recommend const whenever possible to ensure nothing has manipulated the variable since creation and let if you need to mutate it at some point (i.e. counters). Both are also block-scoped, eliminating guesswork and hoisting.

    var total = 0;
    var arrayLength = res.length;
      for (var i = 0; i < res.length; i++) {
        total = total + parseInt(res[i]);
      }
      return total; 

You can use array.reduce to accumulate values. It iterates through an array and uses the callback return value as first argument of the next call. An initial value can be supplied, otherwise iteration starts at the second value with first value as initial value.

Also, always check for NaN when converting strings to numbers. NaN passes through calculations, NaNing everything along the way. It's annoying to track the source of NaN and it may be very far from where it's spotted. Guarding near wherever it potentially comes up is a good practice.

Updating your logic:

function add(stringNumbers){
  const numbers = stringNumbers
    .replace(/(\r\n|\n|\r)/gm, ',') // Normalize delimiter
    .split(',')                     // Split values
    .map(n => parseInt(n, 10))      // Convert to integers

  // Check if numbers are ok. There's probably better APIs
  // to check if something in the array is invalid.
  if(numbers.some(n => Number.isNan(n))) throw new Error('Not a number')
  if(numbers.some(n => n < 0)) throw new Error('Negatives not allowed')

  // Add them all up
  return numbers.reduce((s, n) => s + n)

}