No, it is not possible to set the transparency of a text box created using the Text tools in ArcMap. Transparency can be set for symbolization, but it is only applicable to layers. However, transparency can be set for text in a text box after the text is converted into an annotation feature class, as described in the following method.
Note: Ensure Data View is enabled in ArcMap, and the Draw toolbar is activated. The map scale must be adjusted accordingly as this affects the reference scale of the text. For more information, refer to ArcMap: Reference scale.
Note: Move the text created to the desired location before converting it into a layer because once converted, the text cannot be moved.
Note: Ensure the text in the text box is active before right-clicking Layers or the title of the data frame.
Note: The Convert Graphics to Features tool can also be accessed by clicking the Drawing menu on the Draw toolbar.
Note: A prompt to add to add the exported data to the map as a layer is displayed. Click Yes to add the layer to Table Of Contents.
A transparent copy of the term is produced in a color that honors the transparency settings.