<select name="months" id="months" >
            <option value="3">3 (минимум)</option>
            <option value="4">4</option>
            <option value="5">5</option>
            
</select>

<input type="button" id="elem" value="test"  name="">
 

document.addEventListener("DOMContentLoaded", function() {
  document.getElementById("elem").onclick = function() {
    alert(document.getElementById("months").value);
  };
});