We present a detailed analysis of the process of voltage driven capture of DNA molecules by nanopores. We show that ionic current generates a nonuniform electric field that acts on both the DNA and on its counterions and that the response of DNA to the electric field is affected by its electroosmotic coupling to the mobile counterions. We calculate the voltage and molecular mass dependence of the radius of capture and of the capture rate in the diffusion limited regime. We argue that electroosmotic flow through the DNA coil is suppressed in the vicinity of the pore and present a tentative estimate of the capture rate in the barrier limited regime.