We experimentally demonstrate in a difference-frequency generation mid-infrared frequency comb source the effect of temporal overlap between pump and signal pulses on the relative intensity noise (RIN) of the idler pulse. When scanning the temporal delay between our 130 fs long signal and pump pulses, we observe a RIN minimum with a 3 dB width of 20 fs delay and a RIN increase of 20 dB in 40 fs delay at the edges of this minimum. We also demonstrate active long-term stabilization of the mid-infrared frequency comb source to the temporal overlap setting corresponding to the lowest RIN operation point by an online RIN detector and active feedback control of the pump-signal pulse delay. This active stabilization setup allows us to dramatically increase the signal-to-noise ratio of mid-infrared absorption spectra.