There is no one simple fix for this issue.
MS has worked long and hard to build OneDrive support into Office and Windows. They
really don't want you to be able to disabled it ! So the final solution will take more than one process. And the specific processes will vary depending on Windows and Office versions <sigh>.
Part of the solution (or work around) is simply to not save your files in locations that OneDrive looks at to "sync" (synchronize) between your computer and the cloud. If you can figure out how to turn off synchronization altogether better
still (I haven't figured out that part yet).
This article tells how to turn off the Personal OneDrive on your business computer. The tip does not work on Windows Home because it does not have gpedit.
http://blogs.msdn.com/b/matt-harrington/archive/2014/04/18/how-to-disable-onedrive-file-syncing.aspx
Note, at the end it has a line to the OneDrive equivalent of this forum. You are more likely to get better answers there, from OneDrive "experts".
Here are a couple of similar tips specific to Office 2013 (may still apply to 2016):
Hide Onedrive Option in Office 2013
http://www.howto-connect.com/how-to-disable-save-to-onedrive-in-ms-office-2013/#comment-19255
5 Tips to Customize “File Save” Page in Microsoft Office 2013 Applications
http://www.askvg.com/5-tips-to-customize-file-save-page-in-microsoft-office-applications/