It depends whether it's casual, smart casual, or formal. I'm a male, and for formal dinners, I normally wear a suit. Like you, I like to dress up (for me that means a suit) for the formal night dinners. For casual, for guys, khakis and a button down sport shirt I think would be fine, since it's a Caribbean cruise. For the smart casual (as Royal Caribbean calls it), I wore khakis and a button down shirt, no tie or jacket.
is a link to Royal Carribean's dress code page. The only thing is that they may turn you away if you're wearing shorts or jeans. I know their website says for smart casual, shirts and ties, but like I said, I was just in khakis and a button down shirt, and I was fine.