In an unfinished attic, the biggest issue is wet insulation. It will soak up and hold moisture, creating an ideal breeding ground for mold and bacteria. Wet insulation resting against a wood subfloor can also lead to rot.
If you find a section of wet insulation, pull it out immediately and run a fan and dehumidifier in the affected area before laying down a replacement section.
Finished attics will have the same carpeting, rug, and flooring issues as the rest of the house. Make sure you remove every wet item and dry all wood flooring as thoroughly and as quickly as possible.
Pro Tip: Set a box fan in the attic entry door and have it pull air from the attic for a few days to speed the drying process.