ATX PSU turns itself off when uploading sketch to Arduino
As multiple comments have said, most ATX supplies require a minimum load on the 5V line for proper regulation, many tutorials use ~10 Ohm resistor or smaller on the 5V rail. 

This is due to the type of SMPS topology, typically an LLC resonant converter, that requires a minimum load for tuning of the resonant circuit within a given frequency range.

Solution: Try putting a constant load on the 5V line, and your issue may go away.