Hemostasis disorders play an important role in the pathogenesis, clinical manifestations, and outcome of COVID-19. First of all, the hemostasis system suffers due to a complicated and severe course of COVID-19. A significant number of COVID-19 patients develop signs of hypercoagulability, thrombocytopenia, and hyperfibrinolysis. Patients with severe COVID-19 have a tendency toward thrombotic complications in the venous and arterial systems, which is the leading cause of death in this disease. Despite the success achieved in the treatment of SARS-CoV-2, the search for new effective anticoagulants, thrombolytics, and fibrinolytics, as well as their optimal dose strategies, continues to be relevant. The wide therapeutic potential of seaweed sulfated polysaccharides (PSs), including anticoagulant, thrombolytic, and fibrinolytic activities, opens up new possibilities for their study in experimental and clinical trials. These natural compounds can be important complementary drugs for the recovery from hemostasis disorders due to their natural origin, safety, and low cost compared to synthetic drugs. In this review, the authors analyze possible pathophysiological mechanisms involved in the hemostasis disorders observed in the pathological progression of COVID-19, and also focus the attention of researchers on seaweed PSs as potential drugs aimed to correction these disorders in COVID-19 patients. Modern literature data on the anticoagulant, antithrombotic, and fibrinolytic activities of seaweed PSs are presented, depending on their structural features (content and position of sulfate groups on the main chain of PSs, molecular weight, monosaccharide composition and type of glycosidic bonds, the degree of PS chain branching, etc.). The mechanisms of PS action on the hemostasis system and the issues of oral bioavailability of PSs, important for their clinical use as oral anticoagulant and antithrombotic agents, are considered. The combination of the anticoagulant, thrombolytic, and fibrinolytic properties, along with low toxicity and relative cheapness of production, open up prospects for the clinical use of PSs as alternative sources of new anticoagulant and antithrombotic compounds. However, further investigation and clinical trials are needed to confirm their efficacy.