The best it to have the original box. This usually comes with styrofoam that fit the shape exactly of the monitor. Within it the monitor is inside a large anti-static bag and I plastic wrap the box for protection in case it gets wet.
As thing would have it, this year I travelled twice with 2 monitors. One was in its original box since still had it. Nothing happened to it. It was labelled on 3 sides over the plastic and 3 sides under the plastic in case it got unwrapped. The airline also affixed fragile labels on 3 sides and had someone carry it to the transport rather than going on the conveyor belt. It survived perfectly 4 flights this way.
The other monitor was an older but still high-end color-calibratable model for which I no longer had the original box. That one went into a hard-shell Samsonite (other brands make them too but I own four of these and they outlasted newer ones I bought later) suitcase. The monitor was first wrapped in bubble wrap (the kind with large, roughly 1" bubbles) and then wrapped in towels. Another time I had it inside a rolled yoga-mat. The monitor arrived intact from all its flights. Make sure the suitcase is full of softish items though and do not leave empty space as that will cause the monitor to shift and it may lose a bit of its protection.
Anecdotally, at one of the stopovers I was asked by customs to come watch them inspect my luggage and they kept asking why I would be travelling with a monitor! I had to explain them it was special and showed colors that others didn't.