Four complexes of essential metal ions, Cu(II) and Ni(II), with the new sulfonamide ligand N-(pyridin-2-ylmethyl)quinoline-8-sulfonamide (HQSMP) were synthesized and physicochemically and structurally characterized. Complex [Cu(QSMP)Cl]n (2) consists of a polymeric chain formed by distorted square pyramidal units. In 2, the sulfonamide ligand acts as a bridge coordinating to one Cu(II) through its three N atoms and to another metal ion via one O atom in the sulfonamido group, while the pentacoordinate complex [Cu(QSMP)(C6H5COO)] (3) presents a highly distorted square pyramidal geometry. Complex [Ni(QSMP)(C6H5COO)(CH3OH)][Ni(QSMP)(CH3COO)(CH3OH)] (4) consists of two mononuclear entities containing different anion coligands, either a benzoate or an acetate group. Both units exhibit a distorted octahedral geometry. The interaction of the complexes with CT-DNA was studied by means of UV-Vis and fluorescence spectroscopy, interestingly revealing that the Ni(II) complex presents the highest affinity towards the nucleic acid. Complexes 1 and 2 are able to cleave DNA. Both compounds show promising nuclease activity at relatively low concentrations by mediating the production of a reactive oxygen species (ROS). The interaction of the four complexes with bovine serum albumin (BSA) was also investigated, showing that the compounds can bind to serum proteins. The antitumor potential of complexes 1 and 2 was evaluated against the A549 lung adenocarcinoma cell line, revealing cytotoxic properties that were both dose- and time-dependent.