For one, his Sharingan would help him see the ckahra color of the strings attached to the real Sasori's hands, and be able to know which one to use his most powerful jutsu on, if he manages to get that far. And for two, that's all you need Sharingan for, is to dodge ever attack that comes your way. If you're untouched then what do you have to worry about? How is Sasori able to function? He must have a brain, and if you have a brain, you can deal brain damage, wether it be from physical, or mental damage
Genjutsu would fall under mental damage.