RNA represents a potential target for new antiviral therapies, which are urgently needed to address public health threats such as the human immunodeficiency virus (HIV). We showed previously that the interaction between the viral Tat protein and the HIV-1 trans-activation response (TAR) RNA was blocked by TB-CP-6.9a. This cyclic peptide was derived from a TAR-binding loop that emerged during lab evolution of a TAR-binding protein (TBP) family. Here we synthesized and characterized a next-generation, cyclic-peptide library based on the TBP scaffold. We sought to identify conserved RNA-binding interactions and the influence of cyclization linkers on RNA binding and antiviral activity. A diverse group of cyclization linkers, encompassing disulfide bonds to bicyclic aromatic staples, was used to restrain the cyclic peptide geometry. Thermodynamic profiling revealed specific arginine-rich sequences with low to submicromolar affinity driven by enthalpic and entropic contributions. The best compounds exhibited no appreciable off-target binding to related molecules, such as BIV TAR and human 7SK RNAs. A specific arginine-to-lysine change in the highest affinity cyclic peptide reduced TAR binding by tenfold, suggesting that TBP-derived cyclic peptides use an arginine-fork motif to recognize the TAR major groove while differentiating the mode of binding from other TAR-targeting molecules. Finally, we showed that HIV infectivity in cell culture was reduced in the presence of cyclic peptides constrained by methylene or naphthalene-based linkers. Our findings provide insight into the molecular determinants required for HIV-1 TAR recognition and antiviral activity. These findings are broadly relevant to the development of antivirals that target RNA molecules.