Importance of constant collector current (BJT)
For most "amplification" applications with a BJT, the collector current does vary.  In fact that is usually the output of a BJT amplifier, although often a resistor is used to convert this varying current to a varying voltage.

BJTs have the property that the collector current is largely independent of the collector voltage over a wide part of the operating range.  This can be useful in making current sources/sinks or to send a current signal to a different part of the circuit that is referenced to a varying voltage from the BJT.  In the current source/sink application the collector current is intended to be constant, but otherwise most of the time it is the collector current that is deliberately varied by the circuit as a function of the input.

I can't guess where you got the strange idea from that in general we don't want collector current to vary.  Provide a reference and maybe we can explain why in that particular case collector current should be constant (if that is even true).

Added in response to link in comment below:

That is just a rough plot of how a BJT responds with collector current as a function of collector to emitter voltage with presumably a constant base current.  It is not showing how a BJT is used in a circuit or makes any comments about what we want a BJT to do.  It is simply telling us what a BJT does.

While that plot is correct enough at the rough level of detail it is showing, it is not saying collector current isn't varied in a circuit or that constant collector current is desirable somehow.  One important characteristic of a BJT it is not showing is how the collector current varies as a function of base current.  The plot is presumably at a single fixed base current, although that is not explicitly stated.

Here is a more illuminating plot (copied from http://www.physics.csbsju.edu/trace/NPN.CC.html):



Each trace on this plot is at a different base current.  In this case the base currents go from 10 &micro;A to 80 &micro;A in steps of 10 &micro;A.  The result from the lowest base current is at the bottom and the highest at the top.

What you should really be doing is flipping this around in your mind.  Given the plots show what a BJT does (whether you think it is desirable or not), how can you use these characteristics in a circuit to achieve a goal?

Start by looking at a simple common emitter amplifier.  There are various configurations a BJT can be used in a circuit, but the common emitter amplifier is probably the most conceptually obvious, performs a clearly useful function, and is used a lot.  Sometimes there is a lot of stuff around the transistor to obscure how it is used in a circuit.  Most of that is usually to set the DC operating point in the right place.  However, the main characteristic of a common emitter amplifier is that the input signal is fed into the base such that is causes small changes in base current.  By the nature of how a BJT works as can be seen in the graph above, this causes larger changes in the collector current.  That is then harnessed to make the output signal in the form required by whatever is downstream.  Often we want a voltage out, so putting a resistor in series with the collector causes a varying voltage accross the resistor as a function of the varying collector current.

You can look up common emitter amplifier yourself to get a lot more detail on this.  Another common BJT topology is the emitter follower.  There is also common base and others that you will see less and that may not be as obvious without more careful thinking.  If you plan to design circuits with BJTs, it's a good idea to learn these, why they work, and what they are good for.