Okay, it looks like Scott has added the display of the aura bars as an option in the layout files. The layout files that comes with nUI are default and will get overwritten whenever an update comes out. But if you find a layout plugin that is close to your visual requirements and includes these files you could adjust those remembering that any update to that plugin will overwrite your changes.
That said here goes.
In your wow game folder you should have a folder structure as follows:
And inside those are a bunch of grouping types that you can customize.
Solo, Party, Raid10, Raid15, Raid20, Raid25, Raid40
I would assume only raiding scenarios are annoying and having them in Solo or Party mode are useful so do the following for each of the Raid group folders. I'll use Raid10 as an example.
1. Open up nUI_UnitSkin_Raid10Target.lua in your preferred text editor.
2. Look for the section starting with ["Aura"] a search should find it as the first match if you start from the top of the page. It is roughly around line 675 to help if you are scrolling down.
3. There are 2 types of Aura sections here. the first is debuffs  and the second is buffs 
4. Within each of these blocks is an options section and inside that is an enabled option. By default they are set to true.
5. Simply set them to false and your targets debuffs and buffs won't appear.
Then save and reload the game if still in there or log back in if logged out. I suggest the latter to be sure it is applied.