Setup failed.
Errors were encountered during installation of redistributable packages.
Please close all open programs and try running setup again.
If problems persist, contact DirectX Developer Support.
Error code: S1023
Solution
The problem seems to be caused because my Windows has a Visual C++ 2010 Redistributable that is newer than the one that this SDK is trying to install. The solution, explained here, is to uninstall all the Visual C++ 2010 Redistributable packages and try installing this SDK.
I found that I had the following two packages:
1 Microsoft Visual C++ 2010 x64 Redistributable - 10.0.40219
2 Microsoft Visual C++ 2010 x86 Redistributable - 10.0.40219
I uninstalled them and the DirectX SDK installed without a problem after that.