What you can do is go under the main menu, select the magic tab and click Active Effects
If you currently have a disease, the effects will show up here in red (connoting a negative effect).
Here are the diseases that you can contract in Skyrim as per the Elder Scrolls Wiki
Ataxia: Lockpicking & Pickpocket become 25% harder. This disease is contracted from traps and Skeevers.
Bone Break Fever: -25 Stamina. Contracted from Traps, Bears.
Brain Rot: -25 Magicka. Contracted from Traps, Hagraven
Rattles: Stamina recovers 50% slower. Contracted from Traps and Chaurus
Rockjoint: 25% less effective with melee weapons. Contracted from Traps, Wolves and Foxes
Sanguinare Vampiris: -25% health, progresses to Vampirism. Contracted from Vampires
Witbane: Magicka recovers 50% slower. Contracted from Traps and Sabrecat.
Lycanthropy: Transformation into a Werewolf.
Hope that helped.