We have carried out Brownian dynamics calculations to investigate the effect of DNA-ion interaction on ion transport in a nanopore. We calculated the self-diffusion coefficient of monovalent ions in the presence of DNA in a nanopore and compared the result with that through an open pore, that is, without the presence of DNA. We find that the self-diffusion coefficient of the co-ions is essentially unaffected by the DNA. The self-diffusion coefficient of the counterions, on the other hand, is significantly reduced depending on the ion concentration. At high ion concentration, around 1 M, the effect of DNA on counterion diffusion is relatively small, causing a slight reduction in counterion diffusion coefficient. At low concentrations of a few millimolar, the effect is much larger, resulting in a reduction in counterion diffusion coefficient by a factor of about 2.5. The variation in the self-diffusion of the counterion is well described by accounting for the contributions from two components: the adsorbed counterions and the free counterions. Detailed dynamics of the DNA-counterion interaction is characterized by the varying length of the transient adsorption time of the counterions to the DNA charge sites and the exchange rate with the environment. This variation in counterion adsorption time is attributed to the ionic electric screening effect, which is in turn determined by the ion concentration.