How do I tap into a power supply's standby wire?
I'd suggest you are either connected to the wrong spot in the wiring harness or (much more likely) are using the Arduino Vin pin instead of powering it from the 5 V pin.   

The +5VSB supply should at minimum be capable of delivering 2 amps. 
Look at the ATX spec 3.3.3, and most modern ATX power supplies exceed this and can deliver 3-4 amps. 

The Arduino needs a minimum of 6 V to power it from the Vin pin, so you can't use that. You need to be connected to the 5 V regulated line or to the USB connector XUSB line and leave Vin unconnected.
The Arduino will never have a current flow over 500 mA (that's the XUSB fuse rating. Please refer to the schematic. 

I'd suggest that the best way for you to connect the Arduino to your ATX +5SB is to hack a USB cable and use the USB connector on the Arduino. That gives you a fused supply line with a removable connector ...much better than cutting into the ATX cabling the way you have.