Monogatari games are websites with basic HTML and CSS rules, so you can style any element with CSS by putting rules in the main.css file found in the style folder.
The best ways to go about this would be to style the text-box element with CSS. You could give it a background image, or you could try your hand at CSS Border Images, but I would recommend restricting the size to the dimensions of an image and just using a background image, and padding, personally. It's easier and does the same job.