How to prevent child element executing onmousedown event

Solution 1:

Check event.target to find out which element was actually clicked on.

var element = "#box";
document.querySelector(element).onmousedown = function(e) {
    if (e.target.id !== "box_child_three") {
        alert("triggered");
    }
};
<div id="box">
    <div id="box_child_one">one</div>
    <div id="box_child_two">two</div>
    <div id="box_child_three">three</div>
</div>

Solution 2:

You need to stopPropagation for the event when element three is clicked so that it doesn't bubble up to the parent (box) element.

document.getElementById('box').addEventListener('click', () => 
  alert('triggered')
);

document.getElementById('box_child_three').addEventListener('click', e => 
  e.stopPropagation()
);
<div id="box">
  <div id="box_child_one">one</div>
  <div id="box_child_two">two</div>
  <div id="box_child_three">three</div>
</div>