Magnetic resonance logging plays a pivotal role in petroleum exploration. The magnet system serves as a crucial component of the magnetic resonance logging sensor. However, current designs of magnet systems often rely on empirical structures and lack theoretical research on optimal configurations. This paper proposes a design method for the magnetic core structure of a single-side magnetic resonance logging sensor based on topology optimization using the variable density method. The design domain of the magnetic core in the two-dimensional cross-section model of the logging probe is divided into N small units with densities ranging from 0 to 1. The moving asymptote method of gradient descent is employed to update the density based on gradient information from the objective function, leading to an iterative approach for achieving an optimal solution. Based on these optimization results, a scaled-down prototype of the magnetic resonance sensor is fabricated for experimental verification. With an external radius of 2.5 cm, this sensor can generate a static magnetic field with intensities ranging from 641.0 G (1 G=10^-4 T) to 1108.6 G and gradients from 303.0 G/cm to 683.6 G/cm within the remote region of interest (ROI). The measured data regarding magnetic field distribution aligns with simulation-based optimization results. Subsequently, water mode measurement experiments are conducted using this designed magnetic resonance logging sensor, yielding a signal-to-noise ratio of 23.5.