understanding charecteristics of transistors
What makes the transistor work as desired? Answer: Two VOLTAGES (Vbe and Vce). Hence, the output current Ic is just the result of these initial voltages. 

1.) That is the reason to keep the voltage Vce constant during measurement of the input characteristics.

2.) For measuring the output characteristics we must keep the input bias constant. Here, we have two alternatives: Ib or Vbe. And both alternatives are in use (data sheets). 

In contrast to former statements, the transistor is a voltage-controlled device. This can be (and has been) proven very often. As an example, explanation of the Early-effect as well as the well-known temperatute coefficient (dIc/dVbe=-2mV/K) are a clear indication of the voltage-control principle. There are numerous circuit examples which can be explained on the voltage control principle only.

Nevertheless, it is common practice to show the output set of curves for Ib=const. Why? Because only in this case the slopes of the different curves meet in the Early voltage, whichis a good description of the output conductances. And - as another advantage - the spacing between the several curves is nearly equal. But note, that this has only practical reasons - and has nothing to do with the physical working principle.

In contrary - keeping Ib=const offers the chance to see how an increase in Vce does act upon the VOLTAGE (resp. the E-field) across the B-E pn junction, which in turn gives a small Ic increase.