There is no way to do it without looping through every element.  You could, of course, write a function to do it for you.
function addEventListenerList(list, event, fn) {
    for (var i = 0, len = list.length; i < len; i++) {
        list[i].addEventListener(event, fn, false);
    }
}
var ar_coins = document.getElementsByClassName('coins');
addEventListenerList(ar_coins, 'dragstart', handleDragStart); 
or a more specialized version:
function addEventListenerByClass(className, event, fn) {
    var list = document.getElementsByClassName(className);
    for (var i = 0, len = list.length; i < len; i++) {
        list[i].addEventListener(event, fn, false);
    }
}
addEventListenerByClass('coins', 'dragstart', handleDragStart); 
And, though you didn't ask about jQuery, this is the kind of stuff that jQuery is particularly good at:
$('.coins').on('dragstart', handleDragStart);