The antennas are al "rubber" (actually an elastomer that is often plastic, not rubber at all) and that degrades from oxidation, from ozone, from heat and light, and from improper manufacturing with the platicizers that keep it flexible slowly outgassing and evaporating. As those migrate out the "rubber" stiffens and they might have left the white film.
The way to protect them is to keep them in a cool dark place, and to keep out oxygen. We used to dust them with plain talc, plain talcum powder not anything fancy and not corn starch--which attracts bugs. The talc is inert and easily rinsed off when you want to use them, or just ignored.
These days ArmorAll (the original white stuff, personally, not the new clear stuff) or something similar is supposed to actually replace any plasticizers that leach out, so that's an alternative to use.
Whichever you do, Ziplock freezer bags are less permeable to oxygen than the regular ones, and way better than the thin off-brands. But a real container (i.e. Tupperware) or a Mason jar (or a saved peanut butter jar) is even better. That's going to keep out oxygen even better.
If you're going to store them really long term? You can buy "oxygen absorber" packs online, and toss in a couple of fresh ones before you seal up the jar. That's done commercially in food bags as well, for the same reason. Those are one-shots, just used once and thrown out when you re-open the container.
Of course if any of the antennas are "two buck" specials from unknown makers, often in China? All bets are off, because the chemicals they use are often unstable in the long term. Cheap is cheap, too cheap is always a cheat.