The clinical outcome of SARS-CoV-2 infections can range from asymptomatic to lethal, and is thought to be crucially shaped by the quality of the immune response which includes antibody titres and affinity for their targets. Using Microfluidic Antibody Affinity Profiling (MAAP), we determined the aggregate affinities and concentrations of anti-SARS-CoV-2 antibodies in plasma samples of 42 seropositive individuals, 23 of whom were confirmed to be SARS-CoV-2-positive by PCR testing. We found that dissociation constants (Kd) of anti-RBD antibodies spanned more than two orders of magnitude from 80 pM to 25 nM, despite having similar antibody concentrations. Individual patients showed progressively higher antibody concentrations but constant Kd values, suggesting that affinities did not mature over time. 33 sera showed affinities higher than that of the CoV2 spike for its ACE2 receptor. Accordingly, addition of seropositive plasma to pre-formed spike-ACE2 receptor complexes led to their dissociation. Finally, we observed that the RBD of HKU1, OC43, and SARS-CoV coronaviruses, but not unrelated control proteins, were able to compete substantially with the RBD of SARS-CoV-2 in solution. Therefore, the affinity of total plasma immunoglobulins to SARS-CoV-2 is an indicator of the quality of the immune response to SARS-CoV-2, and may help select the most efficacious samples for therapeutic plasmapheresis.