Probiotics can make illnesses less severe. It can prevent some bacterial invasions as well (or the "take over" aspect of them). But most of the "stomach bugs" passed among kids are viral.
Handwashing is the best thing for preventing stomach bugs. Second best would be an alcohol based hand sanitizer. Alcohol based ones don't produce resistant bacteria so no worries about that.
Hand washing, sanitizer if hand washing isn't available. Probiotics can reduce severity of at least some of the viral (like rotavirus). The bacterial ones are less common for a kid not an antibiotics to pick up.