In the Node view, clear any backdrop or node selections.
Alt+ click on the title bar of a backdrop to only select the backdrop, without the nodes.
If you want to delete the backdrop and all its nodes, simply select the backdrop without using the Alt key.
If you had a node selection, depending on your project, the Confirm Delete window may display. Deselect the Delete Drawing Files and Element Folders. If you have multiple version of a scene, you will want to keep all the files and folders associated with it. If you select this option, all the drawings and folders linked to this backdrop will be deleted.