How can I safely store or dry wet sugar? I recently transferred a batch of home-made vanilla extract to a smaller bottle. I shook out the vanilla beans left in the original bottle into a container with white granulated sugar as there was a lot of extract and seeds left on them. The extract is vodka-based and contains about 35% ABV.

The end result is that this container of sugar is completely soggy and wet now, and I'm wondering how I can safely store it. From what I've read, dry sugar has an indefinite shelf-life because it is hygroscopic, creating a dry environment where nothing bad can grow. This is obviously no longer the case here. High sugar content seems to play a role in the shelf life of fruit preserves but the bigger factor there is probably sterilisation before canning.

How can I safely dry out this wet sugar mixture? 

Should I toss it if it's been sitting at room temperature (with a fine mesh top) for 24 hours? 

Is storing sugar when wet a Really Bad Thing?

Will the alcohol inhibit the growth of anything that might survive the high sugar content?
You shouldn't have any problem. Bacteria need sufficient water to survive, a few drops wetting the sugar are not sufficient for them. As far as I'm aware, even old-style jam (1 part fruit to 1 part sugar, boil some of the water out) is shelf-stable. 

Even if you are close to the limit, it's still not too concerning. The good news is that, when you preserve something with sugar, and get a bit too low on the sugar, it's not bacteria that colonizes it first, it's mold. The mold can be toxic on its own - but it's nicely visible. 

So, as long as you don't have a sticky mass and see mold growing on it, the sugar should be safe. It can be hard to work with though - we have an old question on getting lumped-together brown sugar out of the container, if you need it.