Respiratory viruses are infectious agents, which can cause pandemics. Although nowadays the danger associated with respiratory viruses continues to be evidenced by the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) as the virus responsible for the current COVID-19 pandemic, other viruses such as SARS-CoV-1, the influenza A and B viruses (IAV and IBV, respectively), and the respiratory syncytial virus (RSV) can lead to globally spread viral diseases. Also, from a biological point of view, most of these viruses can cause an organ-damaging hyperinflammatory response known as the cytokine storm (CS). Computational approaches constitute an essential component of modern drug development campaigns, and therefore, they have the potential to accelerate the discovery of chemicals able to simultaneously inhibit multiple molecular and nonmolecular targets. We report here the first multicondition model based on quantitative structure-activity relationships and an artificial neural network (mtc-QSAR-ANN) for the virtual design and prediction of molecules with dual pan-antiviral and anti-CS profiles. Our mtc-QSAR-ANN model exhibited an accuracy higher than 80%. By interpreting the different descriptors present in the mtc-QSAR-ANN model, we could retrieve several molecular fragments whose assembly led to new molecules with drug-like properties and predicted pan-antiviral and anti-CS activities.