Is there a way in InDesign CS6 to add space between the text and its border? Or do I have to just make a shape behind the text instead?
ctrl+shift+b (text frame options dialog), then set your desired text inset
In the "Text Frame Options" dialog box you can adjust the "Inset Spacing" for a text box.
In InDesign CC 2014 on Mac you can find this dialog box by going to Object > Text Frame Options or using the shortcut command+b.