Why are aminoglycoside antibiotics particularly effective for aerobic gram negative bacteria?
Anaerobiosis is incompatible  with effective intracellular accumulation of aminoglycoside antibiotics such as streptomycin and gentamycin.


After an initial binding step involving outer membrane lipopolysaccharide the kinetics of the uptake of aminoglycosides exhibit two energy-dependent processes (EDP I AND II). EDP I  is the slow, rate-limiting stage and is dependent upon the membrane potential ΔΨ, but the details of the process are unclear. EDP II is more rapid and may be triggered by membrane changes arising as a result of translation errors caused by the antibiotic accumulated in EDP I since protein synthesis inhibitors block the transition from EDP I to EDP II.

EDP I requires  a threshold value of ΔΨ, and there is evidence that this threshold often cannot be achieved under anaerobic conditions.

This summary is based on information in this review from 1987.  I haven't been able to find any more recent information.