The best advice is indeed to leave anything precious which mildews either back on the mainland where it has been fine forever, or keep it in a secure climate-controlled environment either there or here. That could be a storage locker, or on the windward side of the Big Island inside a very-expensive-to-operate house.
Here on the east side of the Big Island we have art up on the walls and so far (two going on three years) have only had trouble with two pieces. For whatever it may be worth, here are the factors which -so far- seem to be working in our favor:
1) Fans are on 24/7 and there are plenty of them, so air is stirred throughout the house all the time. We were told by several people (notably Chris at Bay Lighting, in Hilo) that this would really help with preventing mold and mildew in art, furniture, and textiles.
2) We have big eyescrews and "foot" pads on the backs of paintings and prints, which hold them out away from the walls so air can circulate behind the artwork as it hangs on the wall.
3) Windows everywhere allow in plenty of light; even though the artwork is not in direct sunlight, it benefits from backscatter of reflected sunlight inasmuch as this helps prevent fungal growth.
4) Windows and doors are open during hot dry days, only, and all windows and doors are always closed at around 4:30pm or 5:30pm (depending on time of year). Closing the windows and doors as soon as the day starts cooling off prevents much moisture from soaking in via humid air overnight.
5) Dehumidifiers and lava lamps are expensive to operate if you are on HELCO power, but if on PV then much more affordable. I have heard of a lava lamp raising the electricity bill by around $60 in one month when left on continuously. We are on PV so switch on bunches of lava lamps from about noon until 4pm on bright afternoons when power levels are at 100%; by banking the "extra" energy in the form of heat in the lamps, when we shut the windows and doors the lamps radiate heat for many hours and do quite a bit toward dehumidifying the house. We run a big dehumidifier (from Sears, it works fine) and lights inside a sealed closet with textiles, from sunup until sundown daily; there is also a ceiling fan on 24/7 and a solartube ceiling sunlight port in that closet as well as a window (which is always closed). So far there has been zero mildew on any clothing -including leather items- in that dehumidified closet.
6) Every six months we wipe down the inside surfaces of all windows and the exteriors of all furniture with Mold Armor (comes in pale green squirt-bottles, available at stores in Hilo).
7) Heater bars (available at Ace Hardware stores) inside display cabinets, kitchen cabinets, and the piano keep mold away inside those closed spaces.
8) Every six months or so, do check carefully with a flashlight behind artwork on walls and bring into strong sunlight for inspection any objects d'arte (carvings, et cetera) -especially if they are in relatively dark and still spots.
The two items which have mildewed despite all that were an acrylic painting and the black cardboard backing behind a framed print. A couple years ago, just a few months after it was put up, I noticed mildew growing in the black paint only on a acrylic-on-canvas. Although this was a drastic measure I took a clean cloth, sprayed Mold Armor on the cloth, and wiped the mildewed areas directly (both from front and back). Surprisingly, since there is bleach of some sort as part of Mold Armor's ingredients, the painting survived and the mildew so far has not returned. Same treatment for the black cardboard backing of the print, plus a day baking in direct full sunlight. I should probably have completely replaced that black cardboard (and will, if it is ever a problem again).
The UV in sunlight kills many problem organisms but is also hard on pigments, so use it carefully.
If you have artwork you love and want to enjoy by having up, then there is another consideration I would offer: have you made digital copies? Newer digital cameras with high pixel density and extreme sensitivity to light and color can make remarkably good likenesses. Displayed on screens (lit from within, as it were) the quality can be close to the original. Nothing lasts forever, no matter how well it is cared for, so making quality digital copies and storing one set in your safe deposit box will insure that in the event of a housefire or some other disaster at least every trace of the artwork will not be lost. When one has more art than wallspace, digital frames which rotate through images can be a useful solution (especially in bathrooms, where it is too humid to risk any originals).
Artwork which is so significant that it has genuine investment value and/or which you intend to pass along to children or grandchildren as a treasured family heirloom probably should stay in a secure climate-controlled environment rather than being risked on display in a non-climate-controlled East Hawai'i home. Since our paintings, prints, and so on are simply art we love and want to see around our home, we risk it and by using the strategies above so far have had few if any problems, nor lost a single item to date.
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Nina Paley's music video "All Creative Work Is Derivative"
http://www.youtube.com/watch?v=jcvd5JZkUXY&list=FLP7Vhr8C1-hS6BNSE3s6sMw&index=57&feature=plpp_video
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