There is a film position specifically dedicated to ensuring continuity called a script supervisor. The job of the supervisor is to ensure that all actions are repeated in each take, that the lines are spoken correctly, and if not, to inform the director of the mistakes, since he is going to be focused elsewhere.
The simplest way to ensure continuity is to record everything, either in a notebook or a computer. "Sally holds red cup in right hand." This way, one can always revisit it if for some reason the everyday antics clouds your memory.
The next best thing to do is let the actor know. Sometimes their hands and body language act independently of them, and merely knowing they were switching hands can quash any continuity issues.
The supervisor should not hesitate to inform the director of issues; however, he should not cause a shot to be cut just because of a hand issue. The greatest take in the world shouldn't be cut short because of Sally holding her mug in the wrong hand.
The bigger issues arise when the same set is used on consecutive days. In instances such as this, pictures should be taken and replicated the next day to ensure continuity remains intact.