Grab a traffic scraping tool like Fiddler and see what's logged when you try to add the basemap. If you're lucky you'll get a clear error, or there'll be a mismatch between the basemap service ArcMap is trying to use and the one your org should be using.
If there's no issues with the data ArcMap's downloading then you're up a creek, 10.5 is wildly out of service. You can try purchasing your own imagery and making your own basemap tile cache, or use this as a cudgel to get your org onto 10.8.1 where you can squeeze out some support for issues like this.
