My teacher had a similar story, but involved breathing at least one molecule that had been exhaled by Julius Ceaser as he said, “Et tu, Brute?”
Anything which respires (animals all the time, and plants some of the time) produces water. Plants, when they photosynthesize, consume water. These processes, like the biological CO[sub]2[/sub] cycle, are balanced (in fact, any imbalance in the one will also be an imbalance in the other, since they involve the same chemical reactions).
Now, the atoms that the water is made of are very old indeed. The oxygen atoms all predate the Sun (and are hence at least 5 billion years old), having been formed in some previous-generation star which died and contributed some of its material to the proto-solar nebula. The hydrogen atoms, meanwhile, were the first things (given a common understanding of “thing”) to form in the Universe, and so date back over 13 billion years.
If you’re seeing an actual drop, it could very well be guttation, as noted above.
However, plants do indeed give off significant amounts of water via transpiration. The water is in the form of vapor, though. Plants draw water from the ground and put it into the air. This has a significant effect on the climate in areas with heavy vegetative growth.
Rather than moisture on plant leaves coming from the air, then, the converse is the case. The moisture in the air can be often be largely due to transpiration from plants.