I assume by "safe" you mean that the pictured storage boxes won't cause harm to your coins. The short answer is: probably safe. For ancient gold coins, pretty much nothing causes damage due to gold's inertness. Silver coins will be toned due to reactions with hydrogen sulfide in the air, but that won't cause the silver coins to deteriorate. Also, unless the coin is suffering from crystallization or horn-silvering, moisture in the air won't affect the coins. Bronze coins can be affected by moisture and out-gassing of the material used in the wood and cloth of the boxes, but I'm not enough of a chemist to provide better information on this topic (although there is at least one chemist who frequents this site; maybe he'll help). But in general, if your bronze coins aren't suffering from bronze disease, you're unlikely to have problems. It's my understanding that air-tight boxes which trap moisture can encourage the growth of mold and mildew on all coins, which is something to be avoided. The above boxes appear unlikely to be air-tight and thus probably won't cause this problem. Others with more direct experience and better knowledge of chemistry will probably chime in here.
