The QCD axion is a particle postulated to exist since the 1970s to explain the strong-CP problem in particle physics. It could also account for all of the observed dark matter in the Universe. The axion resonant interaction detection experiment (ARIADNE) intends to detect the QCD axion by sensing the fictitious 'magnetic field' created by its coupling to spin. Short-range axion-mediated interactions can occur between a sample of laser-polarized He-3 nuclear spins and an unpolarized source-mass sprocket. The experiment must be sensitive to magnetic fields below the 10(-19) T level to achieve its design sensitivity, necessitating tight control of the experiment's magnetic environment. We describe a method for controlling three aspects of that environment which would otherwise limit the experimental sensitivity. Firstly, a system of superconducting magnetic shielding is described to screen ordinary magnetic noise from the sample volume at the 10(8) level, which should be sufficient to reduce the contribution of Johnson noise in the sprocket-shaped source mass, expected to be at the 10(-12) T/root Hz level, to below the threshold for signal detection. Secondly, a method for reducing magnetic field gradients within the sample up to 10(2) times is described, using a simple and cost-effective design geometry. Thirdly, a novel coil design is introduced which allows the generation of fields similar to those produced by Helmholtz coils in regions directly abutting superconducting boundaries. This method allows the nuclear Larmor frequency of the sample to be tuned to match the axion field modulation frequency set by the sprocket rotation. Finally, we experimentally investigate the magnetic shielding factor of sputtered thin-film superconducting niobium on quartz substrates for various geometries and film thicknesses relevant for the ARIADNE axion experiment using SQUID magnetometry. The methods may be generally useful for magnetic field control near superconducting boundaries in other experiments where similar considerations apply.