Humidity will make a difference, with no doubt whatsoever.
I worked for much of my life as a boatbuilder and can assure everyone here that timber (and tho' cane is botanically grass, structurally you can regard it as timber) does all sorts of interesting things when its moisture content is altered - shrinking, swelling, warping, bending... it's something you really need to know in the boatbuilding trade (boats... water... obvious, really!)
A really dry atmosphere will cause timber to shrink, so much so that wooden boats may leak upon launching until they 'take up'. Some species of timber you can almost tie knots in if you steam it for a period... others - e.g. teak, jarrah - refuse to play.
If you have a dry plank of, say, oak, elm or pine, it will be much stiffer than if you steam it for half an hour or so (depending on thickness) or just soak it for a day or two. Just picking it up and flexing it in your hands and the difference is obvious. A plank is rather the same shape as a reed... just a lot bigger.
So, yes, the ambient humidity will affect your reeds, and soaking them will make increase the amount that they flex for a given pressure.
Whether or not you need to soak your reeds, and for how long, depends on a lot of variables... and a spell of very dry or very humid weather is going to vary the variables (invariably!)
The only answer is to be aware of the problem and...play it by ear!