I can tell you from my own experience that I shared a can of soda with my girlfriend when we were both 8 yrs old when she had a lesion on her lip. My father pulled me aside and told me not to ever do that again but it was too late. Three weeks later I got my first cold sore.
To answer your question, YES you can spread HSV 1 by sharing cups and utensils while you have a breakout! You may even shed the virus while you have no symptoms, but that is unlikely.
The risk of transmitting oral herpes from just sharing a drink with a child is too low to worry about. If you have an active cold sore avoid it until it's gone but in general it's very low risk once a child is of school age - even under that it's low risk as long as they are 6 months or so.
grace