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The resistor is connected to the base of a small signal transistor. So its base current will be of the order of uA. A 250mw resistor is sufficeint for that and for 1K resistor 500mW will be ok.
Remember, in any circuit, even if you use high power rated resistor than required, it won't affect the circuit operation.
Reactive components like capaitors and inductors do not dissipate power. Thats why they are not rated in terms of power. Capacitor has max. voltage rating which should be roughly 1.5 times the supply voltage.
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