We present an occultation study of compact radio sources by the plasma tail of interstellar comet 2I/Borisov (C/2019 Q4) both pre- and near-perihelion using the Arecibo and Green Bank radio telescopes. The interplanetary scintillation technique was used to probe the plasma tail at the P band (302–352 MHz), 820 MHz, and the L band (1120–1730 MHz). The presence and absence of scintillation at different perpendicular distances from the central axis of the plasma tail suggests a narrow tail of less than 6′ at a distance of ∼10′ (∼106 km) from the comet nucleus. Data recorded during the occultation of B1019+083 on 2019 October 31 with the Arecibo Telescope covered the width of the plasma tail from its outer region to the central axis. The systematic increase in scintillation during the occultation provides the plasma properties associated with the tail when the comet was at its pre-perihelion phase. The excess level of L-band scintillation indicates a plasma density enhancement of ∼15–20 times that of the background solar wind. The evolving shape of the observed scintillation power spectra across the tail from its edge to the central axis suggests a density spectrum flatter than Kolmogorov and that the plasma density irregularity scales present in the tail range between 10 and 700 km. The discovery of a high-frequency spectral excess corresponding to irregularity scales much smaller than the Fresnel scale suggests the presence of small-scale density structures in the plasma tail, likely caused by interaction between the solar wind and the plasma environment formed by the comet.