Joe, the "perspiration" is called guttation. Guttation is basically water in the plant that normally would be evaporating through the leaves and has no place to go. Since there is no evaporation (or little) taking place the water must escape so it is being exuded by glands on the leaf blade.

You can cut the plant off near the base which is known as the stem. Although most people call the support of a leaf a "stem" that is not correct since the blade support is a petiole and is technically a part of the leaf.

In time the plant will begin to grow again from the stem. That will not likely take a long time but it will regrow slowly so you'll not have the problem of guttation dripping on your carpet even though you will have a smaller plant in the meantime.

An alternative would be to move the pot into a garage that won't freeze and the plant will very likely go dormant. Once the plant goes "underground" you can cover the pot and wait until spring to bring it back into brighter light. The plant will almost certainly come back in the spring.

Here is a link that might be useful: stem

Sorry, I failed to give better instructions.

Don't cut the plant right at the stem. Instead, just cut off most of the petioles and leave 12 to 15 inches of the plant above ground. If it chooses to go dormant that will not be a problem but if you leave it in your home the chances are new leaves will begin to grow in short order.

I'm with exoticrainforest. I cut my alocasias off above the stem, store them in the basement, water very sparingly throughout the winter (maybe 3-4 times all winter) and have had great success.

I cut off all of the leaf stems close to the trunk EXCEPT the newest, middle leaf. That is where your knew growth comes from. You can cut off just the leaf part of the middle leaf and leave the part where the new leaf will grow. I put mine in the greenhouse for winter but I cut back all of the leaves to make more room and so other plants get more light.

If I have an alocasia potted plant and the leaves are turning splotchy brown and black, what should I do?