My shop is in an unheated section of my basement, so never gets warmer than about 64F, but can be humid in the summer, especially when I'm turning lots of green wood. I use Starbond CA.
I keep a bottle of thin CA, the one I use most, on the bench top, uncapped with one of the microtips on it. When I remember, I clean out the tip after using it like Leo and Dave describe above. Occasionally I have to replace the microtip or cut off the end if it plugs. But the bottle itself will last > 2 years with no problem.
I've got a bunch of other grades of CA - thicker viscosities, some with colors - that I don't use as often. I keep them in jars with some silica gel desiccant so that the air inside the jar is very dry. Moisture is involved in the curing mechanism for CA glues, so it seems to me like this should help. I have bottles that are several years old that still flow and cure fine. I use empty peanut butter containers. Sam's Club and Costco sell Jiffy peanut butter in extra large sizes. The jars are tall enough that you can store a bottle with the microtip still on it.