The best bet would be to use a simple curling iron or hair dryer that has no electronic or temperature control device. This would be the type that has a simple sliding switch with maybe just a high or low setting. This type of device usually has no electronics in them at all Any device with an LED screen or push button switch probably has unprotected electronic components and can be sensitive to power variations.
There is nothing wrong with the power on cruise ships it is just not as clean as the power we get on land from the power grids supplied by our utility companies. Power that is not clean can play havoc with electronic devices. This generally is not a problem for most of our electronic devices as they usually have some way of dealing with this. Most devices the power is supplied through a charging transformer which converts the power from AC to DC which alleviates the issues. Generally anything with a plug-in transformer should be fine.