There is a variety of methods that can be used. The basic idea is of course always that the Picaxe should not see when the supply voltage collapses because the motor suddenly draws a lot of power.
(1) use a separate power supply for the Picaxe. That's a very common approach for robots. E.g. run the motor from a 12V battery pack, and the Picaxe from a separate 6V pack. Also has the advantage that you can use more powerful motors because you are no longer limited by the 5V supply that the Picaxe needs.
(2) Run both from the same pack (e.g. 12V), but use a linear regulator (e.g. 7805) to reduce the voltage for the Picaxe to 5V. Cheaper because only one battery pack is needed. Since the regulator's output voltage does not change when the input voltage changes (at least as long as the input is sufficiently higher than the output), the Picaxe does not care if the motor draws the 12V supply down to e.g. 10V.
In both cases, also provide good power decoupling, i.e. capacitors between power and ground, to buffer sudden spikes in current. At the very least a 100nF (nano-Farad) ceramic capacitor very close to the Picaxe and another one very close to the motor supply points, plus a 10 uF (micro-Farad) electrolytic capacitor for the Picaxe and a large one (1000 uF or more) for the motor. The large capacitors take care of longer spikes, but can't react very fast, the small ones take care of short spikes.
(3) quick fix if you don't have a regulator on hand AND the Picaxe does not draw more than a few mA of current: Put a 100 Ohm resistor in front of the Picaxe's power supply pin, i.e. have the power supply current for the Picaxe (not the motor) flow through this resistor. Then place a 100 uF capacitor as well as a 100nF capacitor between Picaxe power pin and ground pin (i.e. also behind the 100 Ohm resistor). The capacitors act as a reservoir whenever the supply voltage drops (when the motor draws a lot of current), and the resistor prevents the motor from sucking the capacitor empty. BTW, that's a frequently used trick in simple audio amplifiers - in this case the preamplifier stages get decoupled from the current-sucking final power amplifier stage.
Wolfgang