(1) Background: Viral respiratory infections cause life-threatening diseases in millions of people worldwide every year. Human coronavirus and several picornaviruses are responsible for worldwide epidemic outbreaks, thus representing a heavy burden to their hosts. In the absence of specific treatments for human viral infections, natural products offer an alternative in terms of innovative drug therapies. (2) Methods: We analyzed the antiviral properties of the leaves and stem bark of the mulberry tree (Morus spp.). We compared the antiviral activity of Morus spp. on enveloped and nonenveloped viral pathogens, such as human coronavirus (HCoV 229E) and different members of the Picornaviridae family—human poliovirus 1, human parechovirus 1 and 3, and human echovirus 11. The antiviral activity of 12 water and water–alcohol plant extracts of the leaves and stem bark of three different species of mulberry—Morus alba var. alba, Morus alba var. rosa, and Morus rubra—were evaluated. We also evaluated the antiviral activities of kuwanon G against HCoV-229E. (3) Results: Our results showed that several extracts reduced the viral titer and cytopathogenic effects (CPE). Leaves’ water-alcohol extracts exhibited maximum antiviral activity on human coronavirus, while stem bark and leaves’ water and water-alcohol extracts were the most effective on picornaviruses. (4) Conclusions: The analysis of the antiviral activities of Morus spp. offer promising applications in antiviral strategies.