Changing default widget class attribute for all widgets

Is there a way to change the default CSS class for widgets ? I am trying to use Bootstrap 3, and form widgets need a class `form-control` to be displayed properly. I can't seem to find a way to do it without writing lots of code...

The issue provides no file, test, or entry point; begin by tracing how widgets are rendered and how their CSS attributes are selected. Done should provide a supported way to set a default class across widgets and cover the Bootstrap 3 form-control use case with tests.

Written by the indexing model from the issue text.