Based upon the schematics you post most answers here have a good amount to them concerning resistor versus zener.
However, to offer a fair and balanced view towards your specific drawings I'll add a little bit of extra information towards current sinking.
What happens in your circuits is that the transistor is set up as an emitter follower. It means it will want to keep the voltage on its emitter at 0.7V below the base voltage if it possibly can. It will throw its saturation/amplification curves into workhorsing that to work.
If you put a fixed voltage across a fixed resistor, this resistor will want a current to flow. So as long as your load and transistor can support the "required" current, the base voltage of the transistor will determine the current through the load.
Given a set range of supply voltage the current into the base will be quite predictable, so you can set it with both a resistor divider and a zener diode with the proper maths.
Why would you choose one over another?
Well, if you use two resistors, the base voltage will be related to the input voltage. If the resistors are both the same value (low enough to ignore the base current), the base voltage will be 5V at 10V supply, but 6V at 12V supply. That sounds like a problem, but in many cases, by choosing the right balance between the resistor divider it can give a desired effect of limiting the current in the load at power-up of a low-power circuit. It can also give a response to incoming voltages that you want, if you have a control voltage of 6V to 60V, for example, you can turn that into a current curve using just a couple of resistors a transistor and a resistor in the emitter path.
Of course having 50 values of resistors laying around is a very common thing, which adds to the usability of the resistor-only circuit, while usually you have to happen to have the right zener diode.
If, in stead, you have a wobbly power supply, but need a stable current, you should probably use a zener diode.
The zener diode does change voltage over different currents, but if you select a zener diode that is specified to be 5.1V at 5mA, you can assume in your calculations that over 3mA to 6mA it will be relatively stable. You can also calculate the voltage drift using the differential resistance mentioned in the datasheet, but I think that's for a different answer at a different time.
So, if you want the base voltage to stay at 5.1V, you select the 5.1V at 5mAzener, if then the supply is 12V, you select a resistor so that 5mA will go through it:
R = V/I = (12V - 5.1V) / 0.005A = 1.38kOhm.
Say that you could get this value exactly (you can round off to 1.2 or 1.5k because of the relative stability of the zener near the chosen set-point), the voltage can go from:
V = 5.1V + (R * Imin) = 5.1V + (1380 * 0.004) = 5.1V + 5.52V = 10.62V
V = 5.1V + (R * Imax) = 5.1V + (1380 * 0.006) = 13.38V
Before you have to start thinking about checking what the zener does at lower or higher current levels, so it adds a lot of stability with respect to the supply voltage, which then makes the current through the load much more stable.
An alternative (that's even more stable) would be simply two standard diodes, like 1N4148, in series to create a fixed voltage of 1.2V to 1.4V (depending on the current range you are looking at). Using those in the forward direction could give you good stability from 0.1mA to 5mA or from 1mA to 10mA, etc. But it's a bit of a black art to many beginning designers to get to the right set-point calculations. Especially since diode datasheets don't always mention all that data, like forward voltage vs low current.