The LED will be turned off when Vce is less than 1V, so you can use the hFE figures for the transistor in question (which are typically specified at Vce = 1V). As the transistor saturates, the forced beta drops.
At 1.0V Vce, the collector current will be (4.5V-1V)/250 ohms = 14mA. Referring to the datasheet, the gain will be, at a minimum, between 60 and 100 over the range of 10-50mA, probably a lot closer to 100 since 14mA is much closer to 10mA than 50mA. So let's use 80. That means a base current of 14mA/80 or 175uA is just adequate to turn the LED off in a guaranteed fashion (assuming room temperature).
The base current will be approximately (4.5V-0.7V)/Rbase since the Vbe is about 0.7V or so with the transistor on. So the maximum Rbase would be 21.7K. In reality we'd use a lower value because hFE drops at temperature extremes and the datasheet values are for 25°C. At -55°C (fig. 15) the hFE is about half, so 10K would be safe.
In order to drive the transistor well into saturation, meaning a collector current approaching 18mA, we can use Ic/Ib of around 10 or 20. Using 20, we would want a base resistor of
4.2K. (The datasheet only guarantees Vce for Ic/Ib =10).
So the given value for the base resistor is certainly in the correct range for the circuit to work (turn off the LED without the transistor necessarily saturating) and are about right at the higher end of where saturation is guaranteed.