Power-on delay using NPN transistor
This is the power-on delay circuit for the MIDI outputs of the Roland A-50:



R15 and R17 set the base current of the PNP transistor (this circuit is designed to supply 25 mA at 5 V). When powering on, C14 needs to be charged before the voltage at the transistor's base is low enough to switch on; the delay is determined by the time constant of C14 and R15. (For 1–2 s, you need a much larger capacitor.) R14 and D1 are optional; they are a path to quickly discharge the capacitor when power is switched off. R18 is optional; it speeds up switching off the transistor by allowing its base to be discharged faster.

This circuit switches the positive power rail. If you have only an NPN transistor, you have to reverse everything to switch the other rail (ground).

In either case, the module probably will not be able to withstand any voltages applied to its input pins when it is not powered.