My colleague @RossH reports in a comment below that Adobe has fixed the problem. Here's what he says:
Adobe has released a fix for this issue. You can now update your Flash plugin from version 26.0.0.126 to version 26.0.0.131 and no longer experience the various problems described here.
https://get.adobe.com/flashplayer/
Below are the problems that I posted about earlier:
The latest version of Adobe Flash Player (26.0.0.126), released June 13, 2017, may cause buttons to be unreactive in web applications such as SAS Customer Intelligence Studio. For example, you cannot select nodes in diagrams. Details are in Problem Note 60627.
SAS Visual Analytics Design and SAS Visual Data Builder are also having problems on this new version of Adobe Flash Player. See details in Problem Note 60624.
This issue might occur on any supported browser, including Microsoft Internet Explorer, Google Chrome, and Mozilla Firefox.
To avoid the issue, do not download and install Adobe Flash Play 26.0.0.126. If you have already done so, download and install an earlier version of Adobe Flash Player.
SAS is actively researching these issues and will update this information accordingly.