Perfect for a christmas holiday

if you're concerned about temps (air & ocean) you need to go further south than the bahamas.
you could look at grand cayman for similiar topography & great snorkleing.
st. lucia & st. john are good for hiking.
if you want budget with tons to do for all ages look at the maya riviera coast. locogringo.com is a useful site. lots of affordable villas!

First, the Bahamas are not really in the Caribbean but rather off the coast of Florida which is why they can be cool in winter.

As Virginia says, you have to go further south to insure you'll have warmer weather for your Christmas getaway.

Second, you really have to give us some idea of your budget in order for us to make any meaningful suggestions. Terms like "affordable" are meaningless without a dollar point of reference. What's "affordable" for one family may seem overly expensive to another. Let us know how much you are budgeting for airfare, accommodations and food and how many people in your party.

Third, Christmas/New years time period is the peak of High Season in the Caribbean so expect to see the most expensive room rates/villa rental rates of the year. This is not a good time to go for the budget minded traveller. Also, many hotels, resorts and rental villas have minimum stay requirements (can vary from 7 to 14 days) so you may have to factor that into your travel/budget plans.

Fourth, it would help if you told us where you were flying from, how long a trip you are planning, whether or not you you want to fly non-stop from your nearest international gateway airport or if you are willing to take multiple connecting flights and/or ferries to reach your intended destination.

The Caribbean encompasses a big area with scores of islands and literally scores of islands hundreds upon hundreds of resorts/hotels and thousands of rental villas/condos - best first to pick one or two islands than most interest you than look for accommodations.

As stated...the Bahamas will definitely NOT be warm enough to enjoy the ocean during Christmas time.
I also think you may want to look into Mexico's Riviera Maya. We have been there several times during Christmas and the weather and water temps have been great.
There are many resorts..big and small. and also condos, etc. You could explore ruins, zip line, hike, etc. in the area.