β is a good guide for how much base current you have to supply.
In your circuit, if you want to put nearly the whole 10V on R2, there will be about 10 mA in the collector. So you take the worst gain (35) and use that.
That means you need 10 mA/35 = 285 µA in the base, so you chose a resistor of 10/.000285 = 35 kΩ, so use 33 kΩ as a standard value.
If you are just turning transistors on and off, it is quite easy as you can just use more current than you need and it works OK.
For analogue circuits, the circuits are arranged so that the gain must be more than the minimum for the transistor (in this case 35). The circuit must work OK even if the gain is a lot larger.