As convenient as it may sound, I don't think it will catch on. Logistically it would require the line to setup an address for each port, and possibly each ship, you could ship to. They would have to create holding areas for each cruise. Now we add more chances for your stuff to get misdirected - USPS/UPS/FedEx, warehousers, then porters.
Then you have to pack and ship in time for your items to arrive at port before you do.
Besides, as Peg pointed out, shipping your suitcase is more than likely more expensive than the baggage fee you will have to pay with your airline.
All I ask is a tall ship and a star to steer her by; John Masefield
Carnival: Glory 2004, Destiny 2008, Splendor 2009, Freedom 2011, Valor 2012, Dream 2013
Celebrity: Summit 2011
Princess: Ruby 2010, 2014, Caribbean 2013, Coral 2014, Regal 2014, 2016, Royal 2017
Star Clippers: Royal Clipper 2015