Dual supermassive black holes (SMBHs) at ~kiloparsec scales are the progenitor population of SMBH mergers and play an important role in understanding the pairing and dynamical evolution of massive black holes in galaxy mergers. Because of the stringent resolution requirement and the apparent rareness of these small-separation pairs, there are scarce observational constraints on this population, with few confirmed dual SMBHs at <10 kpc separations at z>1. Here we present results from a pilot search for kiloparsec-scale dual quasars selected with Gaia Data release 2 (DR2) astrometry and followed up with Hubble Space Telescope (HST) Wide Field Camera 3 dual-band (F475W and F814W) snapshot imaging. Our targets are quasars primarily selected with the varstrometry technique, i.e., light centroid jitter caused by asynchronous variability from both members in an unresolved quasar pair, supplemented by subarcsecond pairs already resolved by Gaia DR2. We find an overall high fraction of HST-resolved pairs among the varstrometry-selected quasars (unresolved in Gaia DR2), ~30%-50%, increasing toward high redshift (~60%-80% at z>1.5). We discuss the nature of the 45 resolved subarcsecond pairs based on HST and supplementary data. A substantial fraction (~40%) of these pairs are likely physical quasar pairs or gravitationally lensed quasars. We also discover a triple quasar candidate and a quadruply lensed quasar, which is among the smallest-separation quadruple lenses. These results provide important guidelines to improve varstrometry selection and follow-up confirmation of ~kiloparsec-scale dual SMBHs at high redshift.