Point-functionalized carbon nanotube field-effect transistors can serve as highly sensitive detectors for biomolecules. With a probe molecule covalently bound to a defect in the nanotube sidewall, two-level random telegraph noise (RTN) in the conductance of the device is observed as a result of a charged target biomolecule binding and unbinding at the defect site. Charge in proximity to the defect modulates the potential (and transmission) of the conductance-limiting barrier created by the defect. In this Letter, we study how these single-molecule electronic sensors are affected by ionic screening. Both charge in proximity to the defect site and buffer concentration are found to affect RTN amplitude in a manner that follows from simple Debye length considerations. RTN amplitude is also dependent on the potential of the electrolyte gate as applied to the reference electrode; at high enough gate potentials, the target DNA is completely repelled and RTN is suppressed.