Speaker embedding based zero-shot Text-to-Speech (TTS) systems enable high-quality speech synthesis for unseen speakers using minimal data. However, these systems are vulnerable to adversarial attacks, where an attacker introduces imperceptible perturbations to the original speaker's audio waveform, leading to synthesized speech sounds like another person. This vulnerability poses significant security risks, including speaker identity spoofing and unauthorized voice manipulation. This paper investigates two primary defense strategies to address these threats: adversarial training and adversarial purification. Adversarial training enhances the model's robustness by integrating adversarial examples during the training process, thereby improving resistance to such attacks. Adversarial purification, on the other hand, employs diffusion probabilistic models to revert adversarially perturbed audio to its clean form. Experimental results demonstrate that these defense mechanisms can significantly reduce the impact of adversarial perturbations, enhancing the security and reliability of speaker embedding based zero-shot TTS systems in adversarial environments.