JSFiddle - React, Tailwind, and code Playground

HTML

<form name="disponibilidad">
    <div class="arrival">	<span>Start</span>

        <label>
            <select name="llegada-dia">
                <option value="1">1</option>
                <option value="2">2</option>
                <option value="3">3</option>
                <option value="4">4</option>
                <option value="5">5</option>
                <option value="6">6</option>
                <option value="7">7</option>
                <option value="8">8</option>
                <option value="9">9</option>
                <option value="10">10</option>
                <option value="11">11</option>
                <option value="12">12</option>
                <option value="13">13</option>
                <option value="14">14</option>
                <option value="15">15</option>
                <option value="16">16</option>
                <option value="17">17</option>
                <option value="18">18</option>
                <option value="19">19</option>
                <option value="20">20</option>
                <option value="21">21</option>
                <option value="22">22</option>
                <option value="23">23</option>
                <option value="24">24</option>
                <option value="25">25</option>
                <option value="26">26</option>
                <option value="27">27</option>
                <option value="28" selected="selected">28</option>
                <option value="29">29</option>
            </select>
        </label>
        <label>
            <select name="llegada-mes">
                <option value="" selected=""></option>
                <option value="0">jan</option>
                <option value="1">feb</option>
                <option value="2">mar</option>
                <option value="3">apr</option>
                <option value="4">may</option>
                <option value="5">jun</option>
                <option value="6">jul</option>
               ...

CSS

* {
    margin: 5px;
}
div span {
    display:block;
}

JavaScript

$(function () {
    months = ['jan', 'feb', 'mar', 'apr', 'may', 'jun', 'jul', 'ago', 'sep', 'oct', 'nov', 'dec'];

    /* Cachear selects */
    var $ld = $('select[name=llegada-dia]');
    var $lm = $('select[name=llegada-mes]');
    var $ly = $('select[name=llegada-ano]');
    var $sd = $('select[name=salida-dia]');
    var $sm = $('select[name=salida-mes]');
    var $sy = $('select[name=salida-ano]');

    //http://stackoverflow.com/a/1184359/297641
    function daysInMonth(month, year) {
        return new Date(year, month, 0).getDate();
    }

    function adjustDates(selMonthDates, $sel) {
        var $options = $sel.find('option');
        var dates = $options.length;
        //append/remove missing dates
        if (dates > selMonthDates) { //remove
            $options.slice(selMonthDates).remove();
        } else { //append
            var dateOptions = [];
            for (var date = dates + 1; date <= selMonthDates; date++) {
                dateOptions.push('<option value="' + date + '">' + date + '</option>');
            }
            $sel.append(dateOptions.join('')); //reduces DOM call
        }
    }

    function resetDates() {
        $lm.val(function (i, v) {
            return (v == '') ? '0' : v;
        });
        $ly.val(function (i, v) {
            return (v == '') ? '2013' : v;
        });
    }

    var paintCals = function (day, month, year) {

        resetDates();

        //adjust start date
        var selMonthDates = daysInMonth((parseInt($lm.val(), 10) + 1), $ly.val());
        adjustDates(selMonthDates, $ld);

        //If current day selection > number of days in selected month then set the day to max allowed day
        if (day > selMonthDates) {
            day = selMonthDates;
            $ld.val(day); //update selection
        }

        //selected start date
        var selectedDate = new Date(year, month, day);

        //next day from start date
        var nextDay = new Date(selectedDate.getTime() + 86400000);

   ...