I’ve been wrestling with the same issue for a while and I just figured it out. It was a pretty absent minded lapse on my part.
Check the short code of the page/post you have the gallery on. By default, you might have copied/pasted the following: [album id=0 template=compact]
Album id=0 shows all galleries and by default is uneditable.
You need to specify the correct album id=# that corresponds to album you are looking to edit/reorder. check the nextgen plugin options “album” for the correct Album id#. If you have only created one album, the id will most likely be 1.
Once you update the settings and refresh the page, the drag-and-drop options should work.
Hopefully this helps