Humidity is very high there and you are right that it can be problematic for some gear. On top of that, transport is tight and the combination of high-heat plus high humidity makes it even harder to carry. I have been there and to several other Amazon reserves in Ecuador and Peru, so I had a few chances to improve my preparation.
- Take everything that you need, including for emergencies, but nothing more.
- Pack light. Find the lightest items you have and buy replacements if yours are bulky. I found it was possible to save 1lb by switching from a towel to a canga (Brazilian rayonne towel) or micro-towel. Pay attention to the weight of your gear too, the backpack itself, boots, etc. It may not be the most stylish but a disposable rain poncho is much lighter than most rain jackets.
- For clothes, I prefer lightweight cotton because it is very breathable and anything else makes me sweat a lot. Some people can handle polyester clothes are lighter and dry faster. If you opt cotton, get the lightest ones you can find because I had some items that did not dry for weeks in the jungle, even being hung outside and away from rain.
- Electronics are particularly vulnerable. Have waterproof bags for every piece of electronics and a few spare in case some get punctured. Moisture absorption packs help but dry them by a fire or lantern regularly for them to keep working.
- Cameras are even more problematic. If you have professional gear with interchangeable lenses, choose a lens and never change it outside and avoid changing it inside. It takes a fraction of a second for the sensor to fog up. Lenses too will fog up often and you'll miss a lot of shots waiting for it to clear up. If you can stick with one weatherproof camera and lens, that will be best. Otherwise, you can use a plastic enclosure which is expensive. Some people improvise with zip-locks and tape.
- Ensure your backpack and any other bag is either waterproof or place it in a dry-bag. Even if it does not rain, getting splashed is almost inevitable. Make sure any food and medical items are in airtight containers and open them as little as possible. Speaking of electronics, power is scarce and recharging is not always available. Carry some spare batteries, a solar charger would be good too (there is usually a clearing near rivers where light falls several hours a day) and a light splitter. It is not uncommon to have to share a few outlets between a large group.
- Make sure that you can secure all your straps, so that they don't get caught on things and don't dangle into mud, water, etc. It is possible to gain over a pound of weight just from water absorbed by your stuff!
- Mosquito nets are generally provided for sleeping areas but there are also some that you can wear during treks, especially if you take a (guided) night jungle walk.
- The jungle is hot, even during the cooler nights. It is unlikely that you will feel cold, so I would not bring any down sleeping bags or jackets. They would never dry, are bulking and heavier than needed (see previous points).
When travelling to the jungle, there is generally a stay before and after in Quito. Many hotels there can store the gear for you if you are staying there. They will usually request you leave things in a locked bag, so bright a bag and lock for that specific purpose.
The Quito Airport also has lockers but, considering it is so far from the city, I would only leave things there if you were not planning to stay anywhere else in Ecuador.