I would cast a vote for coarse mulch, like 1-2 inch bark chunks.
Fine mulch may absorb too much water and keep it away from the soil and roots. Also, weed seeds can blow onto fine mulch and germinate just fine.
A too-thick layer of very fine mulch (like 2-3" of fine sawdust or coffee grounds) not only absorb too much water, but may even slow down the exchange of oxygen and CO2 into and out of the soil.
Also, if the bottom layer of fine mulch goes at all anaerobic, it may start to ferment instead of breaking down aerobically, and release fermentation products like organic acids and alcohols. Bad for root hairs!