Treating rapidly evolving pathogenic diseases such as COVID-19 requires a therapeutic approach that accommodates the emergence of viral variants over time. Our machine learning (ML)-guided sequence design platform combines high-throughput experiments with ML to generate highly diverse single-domain antibodies (VHHs) that bind and neutralize SARS-CoV-1 and SARS-CoV-2. Crucially, the model, trained using binding data against early SARS-CoV variants, accurately captures the relationship between VHH sequence and binding activity across a broad swathe of sequence space. We discover ML-designed VHHs that exhibit considerable cross-reactivity and successfully neutralize targets not seen during training, including the Delta and Omicron BA.1 variants of SARS-CoV-2. Our ML-designed VHHs include thousands of variants 4-15 mutations from the parent sequence with significantly improved activity, demonstrating that ML-guided sequence design can successfully navigate vast regions of sequence space to unlock and future-proof potential therapeutics against rapidly evolving pathogens.