(1) Background: We aimed to evaluate the association between sociodemographic factors and mental health problems and the sequelae of stroke in South Korea by analyzing the annual Korea National Health and Nutrition Examination Surveys (KNHANES) conducted from 2016 to 2019. (2) Methods: Data were obtained from 32,379 participants who participated in the KNHANES (2016–2019). A total of 567 participants diagnosed with stroke were included in this study. Patients were divided into two groups based on the presence of sequelae: (a) stroke patients with sequelae (n = 227, 40.0%) and (b) stroke patients without sequelae (n = 340, 60.0%). (3) Results: Compared to stroke patients without sequelae, those with sequelae were significantly associated with sex (male, 61.2% vs. 47.6%, p = 0.002), household income (lower half, 78.9% vs. 67.4%, p = 0.005), owning a house (60.4% vs. 68.5%, p = 0.048), marital status (unmarried, 7.05% vs. 1.76%, p < 0.001), depression (13.2% vs. 7.35%, p = 0.045), suicidal ideation (6.17% vs. 3.24%, p = 0.010), and suicide attempts (2.64% vs. 0.88%, p = 0.012). (4) Conclusions: Our study showed that poor sociodemographic factors and mental health problems were significantly associated with sequelae from stroke. Clinical physicians should therefore carefully screen for depression and suicidality in stroke patients with sequelae, especially in those with poor sociodemographic factors.