I collected them last March and stored them in a shed, same temperature as outdoors, covered by mesh (actually old nylon stockings). I checked periodically, except when I was away for over a month. It didn't occur to me to worry about humidity, but perhaps you are right; they get occasionally wet with rain outdoors.
I learned the little I know here and using the other links on the Eurosta solidaginis info page. I was surprised to find perhaps more Mordellistena than flies and also some braconids that I have no idea what they are. I have been thinking about asking Dr. Warren Abrahamson, the author of that site.