They're going to store your bags in closets up on the top deck. You can see them here - even see luggage in one here:
http://www.azurerealm.com/albums/Turks-and-Caicos-Explorer/topdeck.jpg
They're good sized but collapsible luggage is certainly preferred. One photog on our boat had one of those huge Pelican cases for his two DSLR rigs - Explorer had nowhere to store it except outside the lounge door - by the end of the week he hadn't made any new friends since several people kicked/tripped over it. No shoes allowed on the boat either. If you have a main/upper deck cabin they also have a corner desk so in a pinch you could move the chair and store a bag under it - there was built in drawers under the bed.
It's a really big boat also so no issue with finding room for your luggage - we had a full boat and there wasn't any.
I buy cheaper rolling duffel bags - they have wheels/handle and a hard plastic bottom. About $40-50 to about $100 on Amazon. ebags.com has a bunch of them. My current one is a tennis bag that I bought locally. It holds all my dive gear and weighs 3-4 lbs. less than my buddy's rolling Akona dive bag. It has a wide, flat outside pocket that I can just cram my XL fins into. BCD, clothes all my other stuff goes inside. The only downside is that it won't stand on it's end when loaded with dive gear so you have to hold the handle in check-in lines.
My buddy regularly gives me something to carry since I'm in the mid-40's with that bag and a airline legal TravelPro hard sided case I carry on. Many flight attendants use a similar bag - it's where I got the idea.
It holds my video gear, regs, mask, computer and 2 changes of clothes. And I have big video gear, an Amphibico housing/HD camcorder. Once I had to gate check the bag quickly so I've learned to keep the laptop/phone/passport in the top pocket - takes a few seconds to remove them. Since it's a hard sided case, everything was fine.
Buy a luggage scale also - cheap $20 investment and you know where you are. Although on T/C Explorer wet gear is less of an issue - they vent the compressor heat directly thru the dive deck and someone had the bright idea of locating the wetsuit rack directly over it. Also you'll be done diving Friday by noon and don't have to be off the boat till Saturday morning - all our stuff was completely dry by then.
Saudio's Trip Reports
With thanks to Mike Southard, this ^^ is almost identical to what we experienced. Except no dolphins and you do have to leave the boat at 9AM on Sat. now. They recently did some airport renovations afaik - it needed it.
One other thing to mention - due to the big cross-section of the ship, she swings alot on the mooring. And quickly. When you're hanging nearby doing a stop you probably can't catch it if it goes by so just wait 5mins. for it to come around again. The moorings are mostly all near islands but the boat swings out closer to the wall at some - the crew knows this. So I often let everybody else go first, then geared up and talked with the DM for a second till he said go. If they have a 2nd crew member on the dive deck recording times, he's got an even better vantage point and can see the deeper blue of the wall also as it approaches. Saves a lot of swimming.
Don't be like me and assume the white hull is the Explorer. Once it was the Aggressor which I learned only after swimming most of the way to it. Thankfully an Explorer crew member picked me up with the dinghy. No shame in
not wearing yourself out...LOL.
Have a good trip, French Cay is phenomenal - among the best diving I've done in the Caribbean. Lush, deep walls, sharks/rays everywhere - at night you can follow the conch trails. Lots of turtles and basketball sized lionfish also. One dive I counted 22 stingrays near the mooring. There also used to be a friendly spotted nurse shark that approaches divers at one site regularly - someone posted on a recent trip report about it also.