Introduction: Patients with neurological diseases frequently experience anxiety that may exacerbate during the COVID-19 pandemic due to social isolation and prevailing conditions in every society. The purpose of this research study was to explore state and trait anxiety among patients with neurological diseases during the period of COVID-19. Material and method: In the present study were enrolled 100 patients with neurological diseases who visited for regular re-evaluation the outpatient clinics of a private hospital in Attica. Data were collected by the method of interview using the State-Trait Anxiety Inventory Scale—STAI, which included patients’ characteristics. The level of statistical significance was set at p < 0.05. Results: Of the 100 participants, 62% were female, while 52% of the sample was over 60 years old. In terms of anxiety, 50% had scores less than 34.5 and 38 (median) on the scale of state and trait anxiety, respectively (range 20–80). These values indicate moderate to low levels of state and trait anxiety. Statistically significantly higher state and trait anxiety was experienced by patients who were less informed about their disease (p = 0.008 and p = 0.014, respectively) and those who wished to receive information about the pandemic via mass media (p = 0.027 and p = 0.035, respectively). Statistically significantly higher trait anxiety was reported by single patients (p = 0.037), and those experiencing intra-family conflicts (p = 0.008), while higher state anxiety was reported by patients who had mobility problems and other illnesses (p = 0.050 and p = 0.044, respectively). Conclusions: The coronavirus disease 2019 (COVID-19) is a public health issue of global interest and a challenge to address anxiety.