Pull up does not work
The "OE" pin doesn't function quite as you think.  It doesn't set all the outputs to "off" as you would hope. Instead it sets all the outputs to "high impedance".  This is like disconnecting all the wires from the outputs that connect to the transistors, leaving the transistors' bases floating, in an undefined state.

You should connect pull-up or pull-down (depending on the default state you want) resistors to all the outputs so when OE is disabled the transistors get pulled into the correct state.



Update:

After discussion this sounds like the Arduino, when powered off, is interfering with the voltage on the OE pin.  The effect is called "back powering" where power flows out of your circuit back into the unpowered circuit, which can damage one or both circuits.  The power flows backwards into the IO pin of the Arduino, then up through the ESD diodes to its internal Vcc bus, then through its internals into GND.

By disconnecting the Arduino's ground you're breaking that circuit, so the power doesn't flow.

The simplest fix is probably to fit a small diode in series with the OE pin:







simulate this circuit &ndash; Schematic created using CircuitLab

Another, perhaps better, way would be to switch the OE using an NPN transistor:







simulate this circuit