Kaposi’s sarcoma-associated herpesvirus (KSHV) can infect a variety of cells and cause malignant tumors. At present, the use of microRNA (miRNA) for anti-KSHV is a promising treatment strategy, but the instability and non-specific uptake of miRNA still limit its use in the treatment of KSHV. In the present study, we constructed a nano-drug delivery system employing chemical grafting and electrostatic adsorption to solve the problems of easy degradation and low cell uptake of miRNA during direct administration. This nano-drug delivery system is to graft 4-carboxyphenylboric acid (PBA) and lauric acid (LA) onto polyethylenimine (PEI) through amidation reaction, and then prepare cationic copolymer nanocarriers (LA-PEI-PBA). The drug-carrying nanocomplex LA-PEI-PBA/miR-34a-5p was formed after further electrostatic adsorption of miR-34a-5p on the carrier and could protect miR-34a-5p from nuclease and serum degradation. Modification of the drug-carrying nanocomplex LA-PEI-PBA/miR-34a-5p by targeted molecule PBA showed effective uptake, increase in the level of miR-34a-5p, and inhibition of cell proliferation and migration in KSHV-infected cells. In addition, the drug-carrying nanocomplex could also significantly reduce the expression of KSHV lytic and latent genes, achieving the purpose of anti-KSHV treatment. In conclusion, these cationic copolymer nanocarriers with PBA targeting possess potential applications in nucleic acid delivery and anti-KSHV therapy.