Depends on which ship you cruise on. Most ship will require you to dress smart casual for dinner. Also most ships have formal nights (depending on how long the cruise is determines how many formal nights they have on board) Formal nights will require men to wear jacket & tie & women in very smart dress. If you decide that you did not want to go formal then you could always go to another resturant for dinner.
We went on Navigator a couple of years ago to the caribbean and my husband does not like to dress up so we went to johnny rockets or the casual rest I am sure that they had an asian fusion bit to it was slightly different (had a curry and a beer) which was really quite good.The only time that we dressed up but no much was when we went to the steakhouse. other people did dress up really nicely at night and went to the big restuarant. So the answer is depends like the other person has said.