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
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3In InDesign CC, the keyboard shortcut is just Ctrl-B (on Windows). – DawnPaladin Mar 23 '16 at 21:19
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.