It depends on whether final payment has been made or not. If final payment has not yet been made, then most of the time a reduction in price can be given for any fare. (Notice the term 'most' as there are always exceptions to every rule.) However, on Carnival, if the price goes down after final payment has been made, then they will only make the change if you have the early saver fare and you'll get onboard credit for any difference.
Princess, as an example, will reprice your cruise and give you a refund if the price goes down after final payment has been made. So the rules are different depending on the cruise line.
Your agent can definitely help with all this and sometimes they can get things you can't get when prices go down.