I don't think there's a direct way to do this as ArcMap allows to set transparency for whole layer.
But there's a workaround: 1. Make a duplicate of your annotation layer and add them both to table of contents. 2. In edit session select all annotations from first layer. 3. In Attributes window select all elements in a tree and then change their symbol properties to disable callout box (you will have only text in first annotation layer) 4. Set desired transparency for second anno layer and move it below first anno layer. The effect will be like that: [ATTACH=CONFIG]21659[/ATTACH]