When you get sick, your body makes antibodies to help fight off the bad stuff. When you breastfeed baby, he gets those antibodies too. In fact, not breastfeeding during this time puts baby at a greater risk of getting sick, so keep up the nursing. If you don't think baby has been exposed to your illness yet (even though he probably has), you can limit his exposure by washing your hands frequently, limiting face-to-face contact (hold off on the kisses), trying not to cough/sneeze near baby, or even wearing a mask over your mouth and nose while you nurse. If you're especially concerned (or just feel too crappy to nurse), you can let your partner feed baby pumped breast milk.
There's no reason to think breastfeeding would make you sicker. The cuddling involved might even make you feel a little better. Do be sure to take care of yourself so you're well enough to take care of baby, and make efforts to keep your milk supply up: Drink plenty of fluids, nurse often, steer clear of meds (like antihistamines) that can lower your milk supply, and sleep when baby sleeps.