If you want to control the appearance of the border yourself, you can place the content gadget in a layout. Then, from the layout's settings, specify a background color for column in which the content gadget appears. The color you specify will appear as the border color. To control the width of the border, set the Padding values to the width you want. Finally, go to the settings for the content gadget and set the background color to something other than transparent (like white, for example).
Some themes come with content gadget styles that display borders, with or without a heading block. You can select one of these styles within the Gadget style dropdown within the Settings panel for the content gadget.