The cruise lines and immigration request that you have both if your id and birth certificate do not match. Do they always check and ask for the marriage license? No they do not check or ask most of the time. If you find someone who does ask and check you may be out of luck getting on the ship. Is it worth taking the chance to not board the ship. It would not hurt to take the marriage license or a certified copy of one it may hurt if you do not have one even though the chances are small you will need it.
With security constantly changing and some rules getting stricter the easiest way to go is to get a passport. You no longer need your birth certificate or a marriage license if you have a passport. The passport is also a great picture ID.