input type="date"

<input id="date" type="date">

<input id="date" type="date" value="2017-06-01">

var dateControl = document.querySelector('input[type="date"]');
dateControl.value = '2017-06-01';

<form>
  <div>
    <label for="bday">Enter your birthday:</label>
    <input type="date" id="bday" name="bday">
  </div>
</form>

<form>
  <div>
    <label for="party">Choose your preferred party date:</label>
    <input type="date" id="party" name="party" min="2017-04-01" max="2017-04-30">
  </div>
</form>

<form>
  <div>
    <label for="party">Choose your preferred party date (required, April 1st to 20th):</label>
    <input type="date" id="party" name="party" min="2017-04-01" max="2017-04-20" required>
    <span class="validity"></span>
  </div>
  <div>
    <input type="submit">
  </div>
</form>

div {
    margin-bottom: 10px;
    display: flex;
    align-items: center;
}

label {
  display: inline-block;
  width: 300px;
}

input:invalid+span:after {
    content: '✖';
    padding-left: 5px;
}

input:valid+span:after {
    content: '✓';
    padding-left: 5px;
}

<form>
  <div>
    <label for="bday">Enter your birthday:</label>
    <input type="date" id="bday" name="bday" required pattern="[0-9]{4}-[0-9]{2}-[0-9]{2}">
    <span class="validity"></span>
  </div>
  <div>
    <input type="submit">
  </div>
</form>

<form>
    <div class="nativeDatePicker">
      <label for="bday">Enter your birthday:</label>
      <input type="date" id="bday" name="bday">
      <span class="validity"></span>
    </div>
    <p class="fallbackLabel">Enter your birthday:</p>
    <div class="fallbackDatePicker">
      <span>
        <label for="day">Day:</label>
        <select id="day" name="day">
        </select>
      </span>
      <span>
        <label for="month">Month:</label>
        <select id="month" name="month">
          <option selected>January</option>
          <option>February</option>
          <option>March</option>
          <option>April</option>
          <option>May</option>
          <option>June</option>
          <option>July</option>
          <option>August</option>
          <option>September</option>
          <option>October</option>
          <option>November</option>
          <option>December</option>
        </select>
      </span>
      <span>
        <label for="year">Year:</label>
        <select id="year" name="year">
        </select>
      </span>
    </div>
</form>

// define variables
var nativePicker = document.querySelector('.nativeDatePicker');
var fallbackPicker = document.querySelector('.fallbackDatePicker');
var fallbackLabel = document.querySelector('.fallbackLabel');

var yearSelect = document.querySelector('#year');
var monthSelect = document.querySelector('#month');
var daySelect = document.querySelector('#day');

// hide fallback initially
fallbackPicker.style.display = 'none';
fallbackLabel.style.display = 'none';

// test whether a new date input falls back to a text input or not
var test = document.createElement('input');
test.type = 'date';
// if it does, run the code inside the if() {} block
if(test.type === 'text') {
  // hide the native picker and show the fallback
  nativePicker.style.display = 'none';
  fallbackPicker.style.display = 'block';
  fallbackLabel.style.display = 'block';

  // populate the days and years dynamically
  // (the months are always the same, therefore hardcoded)
  populateDays(monthSelect.value);
  populateYears();
}

function populateDays(month) {
  // delete the current set of <option> elements out of the
  // day <select>, ready for the next set to be injected
  while(daySelect.firstChild){
    daySelect.removeChild(daySelect.firstChild);
  }

  // Create variable to hold new number of days to inject
  var dayNum;

  // 31 or 30 days?
  if(month === 'January' | month === 'March' | month === 'May' | month === 'July' | month === 'August' | month === 'October' | month === 'December') {
    dayNum = 31;
  } else if(month === 'April' | month === 'June' | month === 'September' | month === 'November') {
    dayNum = 30;
  } else {
  // If month is February, calculate whether it is a leap year or not
    var year = yearSelect.value;
    (year - 2016) % 4 === 0 ? dayNum = 29 : dayNum = 28;
  }

  // inject the right number of new <option> elements into the day <select>
  for(i = 1; i <= dayNum; i++) {
    var option = document.createElement('option');
    option.textContent = i;
    daySelect.appendChild(option);
  }

  // if previous day has already been set, set daySelect's value
  // to that day, to avoid the day jumping back to 1 when you
  // change the year
  if(previousDay) {
    daySelect.value = previousDay;

    // If the previous day was set to a high number, say 31, and then
    // you chose a month with less total days in it (e.g. February),
    // this part of the code ensures that the highest day available
    // is selected, rather than showing a blank daySelect
    if(daySelect.value === "") {
      daySelect.value = previousDay - 1;
    }

    if(daySelect.value === "") {
      daySelect.value = previousDay - 2;
    }

    if(daySelect.value === "") {
      daySelect.value = previousDay - 3;
    }
  }
}

function populateYears() {
  // get this year as a number
  var date = new Date();
  var year = date.getFullYear();

  // Make this year, and the 100 years before it available in the year <select>
  for(var i = 0; i <= 100; i++) {
    var option = document.createElement('option');
    option.textContent = year-i;
    yearSelect.appendChild(option);
  }
}

// when the month or year <select> values are changed, rerun populateDays()
// in case the change affected the number of available days
yearSelect.onchange = function() {
  populateDays(monthSelect.value);
}

monthSelect.onchange = function() {
  populateDays(monthSelect.value);
}

//preserve day selection
var previousDay;

// update what day has been set to previously
// see end of populateDays() for usage
daySelect.onchange = function() {
  previousDay = daySelect.value;
}