DNA can stabilize silver nanoclusters (Ag_N-DNAs) whose atomic sizes and diverse fluorescence colors are selected by nucleobase sequence. These programmable nanoclusters hold promise for sensing, bioimaging, and nanophononics. However, DNA's vast sequence space challenges the design and discovery of Ag_N-DNAs with tailored properties. In particular, Ag_N-DNAs with bright near-infrared luminescence above 800 nm remain rare, placing limits on their applications for bioimaging in the tissue transparency windows. Here, we present a design method for near-infrared emissive Ag_N-DNAs. By combining high-throughput experimentation and machine learning with fundamental information from Ag_N-DNA crystal structures, we distill the salient DNA sequence features that determine Ag_N-DNA color, for the entire known spectral range of these nanoclusters. A succinct set of nucleobase staple features are predictive of Ag_N-DNA color. By representing DNA sequences in terms of these motifs, our machine learning models increase the design success for near-infrared emissive Ag_N-DNAs by 12.3 times as compared to training data, nearly doubling the number of known Ag_N-DNAs with bright near-infrared luminescence above 800 nm. These results demonstrate how incorporating known structure-property relationships into machine learning models can enhance materials study and design, even for sparse and imbalanced training data.