Don't store under glass
Bobbins should be stored out on a shelf or on a stand, putting them under glass, where there is little or no air circulating will cause blooming
Not just wood, but also bone bobbins
Remember, pewter inlay appears on both wooden and bone antique bobbins.
A Nasty History
Lead has a toxic history (sorry, couldn't resist the pun), from being included in the paint on toys being used as a sweetener in drinks. Yes, there was a history of adding lead to cider to make it sweeter leading to brain damage and death and is the origin of the jokes about 'country yokels'.
Lead can oxidise and start to have a white 'bloom' on it. This is a sign of 'lead disease'.
Keeping antique bobbins in airtight containers will cause blooming.
The first thing is do not touch this with your bare hands. Put on gloves.
If you find that an antique bobbin has developed blooming then follow the advice from the Western Australian Museum on treating lead bloom.