The effects of plasma density and laser energy on the stability of laser produced monoenergetic electron beams are investigated. Fluctuations in the principal beam parameters, namely, electron energy, energy-spread, charge, and pointing, are demonstrated to be minimized at low densities. This improvement in stability is attributed to the reduced time for pulse evolution required before self-injection occurs; i.e., that the pulse is closest to the matched conditions for these densities. It is also observed that electrons are only consistently produced above a density-dependent energy threshold. These observations are consistent with there being a threshold intensity (a0≳3)(a0≳3) required for the occurrence of self-injection after accounting for pulse compression.