By badge do you mean logo?
If so, follow these steps:
Go to your Dashboard > Appearance > Widgets
Add a text widget to your sidebar
Place in the following code (leaving the “”, <>, etc intact but filling it out with your own information)
<a href="othersiteurl"><img src="logolink" /></a>
Here’s an example that you can paste in so you can see what I mean… <a href="http://seaworthygirl.com/"><img src="http://seaworthygirl.com/wp-content/uploads/2012/06/seaworthygirlfullogo400.png" /></a>