Measuring the power output of a hand-crank generator
The power should not drop proportionally with respect to the size (resistance) of the resistor (load).  But it will probably drop none the less because of how people are made.

Consider halving the resistance from 1K ohms to 500 ohms.  Using this equation it would appear the voltage would be halved as well:

V = I x R


But the lower resistance will allow more current to pass through the load.  

Now the question can be asked if the power source is a constant voltage or a constant current type?  If a constant voltage the power source will compensate for a lower resistance and supply more current.  So the power would increase.  The power equation is:

P = V x I


Your version of it combines both the above equations:

V = I x R 
rearrange: 
I = V / R 
substitute: 
P = V x (V / R) 
simplify: 
P = V^2 / R


So the power would double.

But for a constant current supply the current would stay the same no matter what the resistance of the load.  Instead the voltage will be reduced by half when the resistance is reduced by half.  Remember:

V = I x R


But a person turning the crank on a generator will likely neither simulate a constant current nor constant voltage source of power.  It is more complex.  

For example, most people can climb a stair well easier than they can climb a rope to the same height.  Even though the person who climbed the stars and the person who climbed the rope end up having the same potential energy. 

Similarly, a person may find they can turn a generator connected to a high resistance load for an extended period of time.  Where as a person may find they can turn a generator connected to a low resistance load for only a short period.  This may give the impression that a lower load resistance effects the power calculations.  When it is likely that people are designed to deliver more power when the demand is spread across longer periods of time.