Though the source HTML of the checkout page can't be modified, you can use the Code Editor to insert Javascript that makes simple modifications once the page is rendered. Here's an example for adding a note to the gift message section:

{% block page_javascript %}
$( document ).ready( function(){
<p style=\"border: 1px solid rgb(224, 224, 224); margin: 0em; padding: .2em; clear: both;\">\
Please add a message for the recipient so we can include a note in their first shipment.\
{% endblock %}

A page can only have one page_javascript block, so if you already have a block with that name in your page, be sure to combine them into one.