Just trying to make sure I've grasped what you're talking about when referring to the sides of a door, some hamfisted ascii art:
Left side viewCode:
Slatted door sides
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Open door sides
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Assuming my visualisation of your open door side concept is correct then my hunch would be that slats are included in commercial designs firstly to prevent fingers/pens/BLT sandwiches getting wodged down there, and secondly for perceived aesthetic benefits.
Slats seen in decent designs like Antec/Fractal etc are likely to be open enough that (a) they don't introduce any impairment or turbulence to airflow (at least at the speed we're likely to run case fans) and (b) don't produce any effective barrier to sound (not that this is necessarily needed in this position).
As such I'd further guess that an open design over loosely slatted would in theory produce better airflow and have no effect (for better or worse) on noise. In practice the theoretical airflow benefit is probably negligible or insignificant.
I suppose that one option might be to leave open the side which will be further away from you when the case is in use and close up the side sitting nearest you, coupled with as dense a layer of sound absorbent material as you can fit - however I'd guess that the likely noise benefit in this scenario to be placebo only.
Disclaimer: you'll have noticed I have frequently used the words "hunch" and "guess". Beyond a physics education (which (a) only tangentially touched upon such issues, (b) has not subsequently been used in work, and (c) is now rather old) I have no qualifications in this regard beyond a persistent grumpiness at all things noisy which occasionally motivates an attempt to fix such irritants.