Watch footage of the Saturn V (or later shuttles and other rockets) and you will notice they are never left standing upright on their own – there is always a gantry structure alongside them to provide support. This was a big steel structure with arms that reached out to clamp the rocket in place and hold it vertically.
When it came time for launch, these arms release at the very last second once the rockets engines are already firing – at this point the rockets engines control the stability of the rocket, as they are designed to move and direct the thrust of the rocket. If the rocket starts tilting to one side the engine directs the thrust the opposite way to counteract that and bring it back to vertical.
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