You may see the following error message after updating to ArcGIS 10.4.
In your My Esri account where you download 10.4, Scroll down until you see a section for "Database Support Files."
There should be two clients to download and install:
- Microsoft SQL Server 2012 SP3 Native Client
- ODBC Driver 11
Install those clients, and you should have connectivity to your databse, again.