I have done it both ways for my wife.
It is easier to check it with the luggage referring to special handling. Make sure you have a cabin baggage tag on it.
Have available the oxygen prescription in case, a copy will do.
You will also need to contact the cruise line that you are bringing the equipment.
There usually are forms needed to send or fax back to special needs. Contact the cruise line and explain about oxygen and have them turn you over to the department that handles it. Keep a record of those you spoke with and get a direct number and extension.
Copy all the forms once filled out and keep it for your records.
Sea of Cortez on Ryndam 9-28-2005
Round trip Los Angeles to Hawaii, Summit March 26, 2006
Vancouver – Alaska – Los Angeles, Sept 16, 2006 on Summit