When I candy-coat apples and let them sit out, the candy coating becomes sticky. I live in the Southern U.S., so humidity is usually pretty high, but other people in my area make them as well.
You may not be getting your candy to the right temperature. If you go too low, the candy never hardens completely. Are you using a candy thermometer?
If they're fine at first, but turn sticky later, you could try storing them in a drier place (don't put them in the refrigerator, of course). You might also want to try rolling them on sugar (if the issue is with the entire coating) or setting them on a layer of sugar (if the issue is presentation), which can help reduce the stickiness (and doesn't look bad).
Consolidating some of the other answers. The problem is that you're either
- Not getting the sugar up to Hard Crack tempeature
- Storing them in too-wet of an environment
- Adding too much of something (i.e. cream) after the sugar has reached temperature, so it's keeping it from getting hard.
If you're not sure about the temperature of your sugar, that's the first place to start. You say only gets sticky when sitting out, which I assume means it wasn't sticky before it was sitting out, so this seems unlikely.
If your environment is just too humid, which sounds pretty likely, there's not a whole lot you can do about that. Maybe try making them on a dry day and storing them in a sealed container until you need them? Commercial confectioners use things like carnauba wax to coat their candies so they resist stickiness in humid environments... not sure if a home-cook equivalent exists.
Edit: just spoke to an old classmate who has made a lot more candy than me— she suggested putting a some food grade mineral oil on a paper towel and lightly wiping the apples down with it right after the shell hardens.
If you're adding something to the sugar after its done cooking, such as cream, use a lower-moisture ingredient (for example, butter instead of sugar) if it's available, or use less of it.