The attacks by the Iran-backed militia in Yemen since November have slowed trade between Asia and Europe and raised concerns about an escalation of the war between Israel and Palestinian Hamas in Gaza.
Royal Caribbean said in a statement on Thursday it had cancelled two voyages so far.
One from Muscat to Dubai was meant to take place Jan. 16-26, and another from Dubai to Mumbai was scheduled for Jan. 26-Feb. 11.
It also amended last week the itinerary of a cruise between Aqaba and Muscat to disembark guests in a port city near Athens.
"Our global security team continues to closely monitor the situation in the region and we will make additional changes if required," Royal Caribbean said.
Swiss-Italian operator MSC Cruises said on Wednesday it had cancelled three trips in April from South Africa and the United Arab Emirates to Europe due to the Red Sea crisis.
"The safety of passengers and crew is the number one priority and as there was no viable alternative itinerary, the company has regrettably had to cancel the voyages," MSC Cruises said. "The three ships will transfer directly to Europe without any passengers on board and avoid transiting through the Red Sea."
Although thousands of passengers are affected, the impact on cruise operators at a global level is not expected to be significant, said Todd Elliott, CEO of Florida-based travel agency Cruise Vacation Outlet.
"This is a small part of their overall fleet and multi-year itineraries so they will be able to overcome this easily," Elliott said.
Italy's Costa Cruises told Reuters on Thursday that routes "remain unchanged" and only two of its cruises scheduled to transit through the Red Sea in March and April could be affected, including the last leg of a round-the-world trip.
Carnival said its global security team was working with global security experts and governments to prioritise safety, including adjusting its itineraries if needed.
