We demonstrate the application of amphiphilic ionic poly(n-butylmethacrylate)-block- poly(2-(dimethylamino)ethyl methacrylate) diblock copolymers (PnBMA40-b-PDMAEMA40, PnBMA40-b-PDMAEMA120, PnBMA70-b-PDMAEMA120) for dispersing multiwalled carbon nanotubes (MWCNTs) in aqueous media, a subsequent efficient surface modification of screen-printed electrodes (SPEs), and the application of the modified SPEs for DNA electrochemistry. Stable and fine aqueous dispersions of MWCNTs were obtained with PnBMAx-b-PDMAEMAy diblock copolymers, regardless of the structure of the copolymer and the amount of MWCNTs in the dispersions. The effect of the diblock copolymer structure was important when the dispersions of MWCNTs were deposited as modifying layers on surfaces of SPEs, resulting in considerable increases of the electroactive surface areas and great acceleration of the electron transfer rate. The SPE/(PnBMAx-b-PDMAEMAy + MWCNT) constructs were further exploited for direct electrochemical oxidation of the guanine (G) and adenine (A) residues in a model salmon sperm double-stranded DNA (dsDNA). Two well-defined irreversible oxidation peaks were observed at about +600 and +900 mV, corresponding to the electrochemical oxidation of G and A residues, respectively. A multi-parametric optimization of dsDNA electrochemistry enables one to get the limits of detection (LOD) as low as 5 μg/mL (0.25 μM) and 1 μg/mL (0.05 μM) for G and A residues, respectively. The achieved sensitivity of DNA assay enables quantification of the A and G residues of dsDNA in the presence of human serum and DNA in isolated human leukocytes.