With increasing performance of deep learning, researchers have employed Deep Neural Networks (DNNs) for wireless communications. In particular, mechanisms for detecting Wi-Fi frames that use DNNs demonstrate their excellent performances in terms of detection accuracy. However, DNNs require significant amount of computation resources. Thus, if the DNN-based mechanisms are used in mobile devices or low-end devices, their battery would be quickly depleted. Spiking Neural Networks (SNNs), which are regarded as the next generation of neural network, have advantages over DNNs: low energy consumption and limited computational complexity. Motivated by these advantages, in this paper, we propose a mechanism to detect a Wi-Fi frame using SNNs and show the feasibility of SNNs for Wi-Fi detection. The mechanism is composed of a preprocessing module for converting collected signals and an SNN module for detecting Wi-Fi frames. The SNN module employs Leaky Integrate-and-Fire (LIF) neurons and Spike-Timing-Dependent Plasticity (STDP) learning rule. To reflect the features of an actual neuromorphic hardware system, our SNN module considers memristive synaptic features such as non-linear weight updates. Our experimental study demonstrates the detection capabilities of the proposed mechanism are comparable to those of previous mechanisms that use DNNs, CNNs and RNNs while consuming definitely less energy than the previous mechanism.