Displaying table results as responsive

I’m using the Responsive theme and displaying information from a database in a table. I’m running into an issue keeping the database table from bleeding off the page when viewed on a smartphone. The table itself is responsive until you really begin to shrink the width of the page. I’ve attempted to apply width code to the table, but that doesn’t fix the issue. Is this more of a CSS edit that would be required in the style.css instead?