human speech noise filter
A speech communication channel as used in telephony typically has a frequency response of 300 Hz to 3 kHz. Although this rejects a lot of the energy in normal speech, intelligibility is still quite good - the main problem seems to be that certain plosive consonants, e.g. "p" and "t", can be a little hard to discriminate without the higher frequency components.

So you're probably looking for a "sweet spot" somewhere between using the full 20 Hz - 20 kHz bandwidth typically found in consumer audio and the most aggressive filtering used for voice comms (see above). I would suggest starting with a bandpass filter from say 50 Hz to 8 kHz. It will probably only improve SNR by a few dB at best, but it may help, particularly if you have a lot of high frequency background noise.