Separate PICAXE Download board Serial In pull down resistor
A link to the documentation of the board would possibly permit a more definitive answer, but general rules of thumb can be applied.

The reason a pull-down resistor is suggested for the serial in pin, is to prevent the input from floating when not driven by anything. If the pin were allowed to float, it would be sensed as a series of random voltage changes due to electromagnetic interference, and the device would attempt to interpret it as some (nonsensical) serial signal. 

To prevent such floating, in general anything from 100 KOhms down to 10 KOhms will be used as a pull-down value. The smaller values will make the input less sensitive to EMI when not being driven, but will also make the output of the driving circuit work harder (more current) to generate a high on that pin. Wasted power, more heat at the driver, faster battery drain (if applicable), not nice. 

Too high a resistance means that relatively small electromagnetic interference, such as from a nearby motor or electronic device, would permit the serial input pin to perceive state changes due to the noise, so the purpose is defeated. 

In general, one would start with 100 K, and if interference persists, work downward in steps to the 10 K shown in the schematic.  The exact value of the pull-down resistor is non-critical.