When I was collecting vintage fabrics to sell, I used brass safety pins to hold on archival (non-acid) light cardboard tags. Never a problem despite living here in the humid Seattle area and storing some of those fabrics for decades.
I often find old fabric at thrift stores and estate sales with this sort of problem, staples do it too.
On a side but related note, for those of us who store our fabric in cardboard boxes, never have those boxes directly on concrete floors, they will wick up small (or large!!) amounts of moisture and give you that musty smell.