You don't indicate your location/climate so I'm not sure which black bamboo you mean nor which pest you may have but...do not cut out any old growth until the new shoots are already making lots of leaves on new culms, after this it is fine to start thinning. I would blast the plants with a hose to try and dislodge the webs rather than cut the culms - you can always cut them later if needed but they are very valuable to a young plant.
Bamboo mites will make the white checkerboard type pattern visible on the top of the leaf, on the bottom of that leaf you 'mite' be able to see some webbing that you can rub off. I do not believe that bamboo mites will make webs at the branch/culm junctions, that is likely a different bug. I don't think mites will cause the brown leaves.
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Brad Salmon, zone 5b/6 Southern Indiana
Winters -20 to -25C. Summers 30 to 35C , humid. 115 cm annual precipitation, frost free from May through early October. 259.3 meters elevation. Growing 150+ species.
http://www.needmorebamboo.com/