I have been on about 20 cruises including to the Bahamas and all over the Caribbean and here what I recommend based on my experiences.
If you have paid for your cruise that will include your cabin, use of all facilities and activities on ship (except spa services like hair, nails and wraps which have a fee), and all meals including breakfast, lunch dinner, and snacks, plus free room service. Included also will be coffee, tea, juices, milk, and any punch or lemonade they may have. You will have to pay for alcoholic beverages and soft drinks. A 15% gratuity is added to every drink order, so you don't pay them as a lump sum but per order.
When you first board the ship the cruise line (whichever one you go on) will set up at check in a cruise account for everything that you buy on board the ship. And that includes both drinks and souvenirs. So all of the alcoholic and soft drinks that you order will go on that tab. Also, if you want to go on any tours from the ship or buy any of the photos of you that they take you will pay for them via the tab. Then at the end of the cruise they will add the gratuity which will be $10 per person per day (for adults but maybe be less for kids). Then before you get off the ship they will charge this to your credit card, OR you can make arrangements to pay cash for everything if that's what you want.
So ships are basically cashless. The only cash you will need is for the casino and when you get off the ship to shop or go on a tour or sightseeing. So if you plan to pay your tab by credit card and not drink a lot you may not need any more than $300 to $400. But if you plan to pay cash for everything then you may need more. Your cabin will have a safe to store cash if you decide to do that.
In the Bahamas and the Caribbean you can usually get a tour or taxi to do the same as ship's tour for much less and the people doing these will be right at the pier, just tell them what you want to do. BUT, be sure that you know exactly what time you must be back on ship and be sure to plan to be back at least half and hour before that time. The one benefit of booking through the ship is they will NEVER leave you at the port, not so if you go on your own.
In the Bahamas there are not really a lot of really interesting attractions, mainly the beach activities. But here's a web site: http://www.bahamas.com/bahamas/island/templrgstandard.aspx?sectionid=14168&level=2
But one of the major attractions there is is the Atlantis hotel which has an aquarium and is a huge property that I understand cruise passengers can also use to some extent. You can get there easily and quickly by taxi. Here's that resorts web site: http://www.atlantis-resort-bahamas.com/
When we cruise we typically buy one or two photos, maybe a few souvenirs, a couple drinks a day and with our gratuities added our tab is usually under $200 each for a 7 day cruise. Then we take a couple hundred to spend in the casino and on the trinkets and stuff that you see in port stops. If you plan to buy big stuff like jewelry use a credit card.
The cruie ship will have an ATM on board near/at the pursers' desk AND here's a tip. If you run low on cash, go to the casino window and request some cash on your cruise card. If/when you do that there are no ATM fees because they are just adding to your on-board account/tab.