After decades, the glycopeptide vancomycin is still the preferred antibiotic against resistant strains of Gram-positive bacteria. Although its clinical use is strictly regulated, the gradual spread of vancomycin-resistant bacteria, such as glycopeptide-resistant and glycopeptide-intermediate Staphylococcus aureus and vancomycin-resistant Enterococcus spp., is a serious health problem. Based on the literature data and previous studies, our main goal was to assess the antimicrobial potential and to study the structure–activity relationship of new eremomycin aminoalkylamides. We designed and synthesized a series of new eremomycin amides in which eremomycin is conjugated with a hydrophobic arylalkyl group via an alkylenediamine spacer, and tested their antibacterial activities on a panel of Gram-positive strains that were sensitive and resistant to a “gold-standard” vancomycin. Based on the data obtained, the structure–activity relationships were investigated, and a lead compound was selected for in-depth testing. Research carried out using an in vivo model of staphylococcus sepsis, acute toxicity studies, and the estimated therapeutic index also showed the advantage of the selected eremomycin amide derivative in particular, as well as the chosen direction in general.