Is it correct to directly modify a supplied Drupal theme? I downloaded and installed the Bootstrap theme for a new Drupal 7.28 site.

Is it correct to simply modify this theme directly to build my site, or should I create a custom theme and somehow "base" it on this theme? 

Put another way, is theming analogous to downloading the Bootstrap CSS and overriding selectors, or directly modifying the CSS itself?
No, it's not correct to directly modify a contrib theme (or Drupal core, or a contrib module). Once you start doing that, you cannot easily update the theme anymore. It a few months, you, or another, developers may forget the theme was modified, do what should be a easy update and break everything.

You have two options to change and existing theme. You can rename the theme before making any other change (ie. fork the theme to a custom theme). You can also create a sub-theme of your chosen theme. The sub-theme can override template and theme functions, and adds ins own process and pre-process functions. But it will inherit all non-overridden templates and theme functions and all process and pre-process functions.