This is a question about the power necessary to control a transistor circuit, i.e. the base current in BJTs vs. the gate current in MOSFETs.
Why is there quiescent power dissipated by base currents of BJT's?
Why doesn't it happen with MOSFETS?
Electronic – Quiescent power of BJT vs MOS transistors
bjtmosfet
Best Answer
The key to the answer is buried in the long name of the MOSFET, which is Metal Oxide Semiconductor Field Effect Transistor.
The oxide layer isolates the gate from the channel (drain-source). For the MOSFET to be controlled, you apply a voltage between gate and source, but because of the isolation, the voltage cannot cause a current to flow from gate to source. Gate currents become significant only when you change the voltage, i.e. when you put more charge into the gate or you withdraw charge from the gate. The resistace being formed between drain and gate depends on the field created by the nearby gate, i.e. the gate voltage. Thus, MOSFETs are essentially voltage-controlled devices, not current controlled devices.
A BJT will allow (and require) current to flow into the base when you apply a voltage between the base and the emitter. This current controls the current from the collector to the emitter. Even if you don't care about the "transistor action", the base-to-emitter part will act like a diode, i.e. it will allow current to flow if biased the right way. One can say that BJTs are current-controlled devices more than they are voltage controlled devices, although you first need a voltage as the cause for the current to flow. Now... Current alone still doesn't explain the power (current * voltage) dissipated by the BJT. Just like with diodes, there is a voltage drop (approx. 0.7 V) across the B-E part of the BJT. Multiply your base current with this "diode drop", and you have a power dissipated by the base current.
This explanation is quite short and by far doesn't cover everything, but it gives you at least some ideas where to start digging further.
However, your question makes sense only when looking at either transistor in the context of a whole circuit. One npn BJT alone can be off with no base and no collector current, so there is no quiescent current. Consider a larger logic circuit in RTL (resistor-transistor logic) vs. CMOS (complementary MOS) logic. With RTL, there will always be some transistors that conduct and some that don't, so some will dissipate power and some won't, even if the circuit "does nothing", i.e. doesn't change its state. CMOS will only require some gate (dis)charge current whenever it changes its state.
Something similar is true for MOSFET-based LDOs (voltage regulators) vs. BJT-based LDOs.