My wife and I are wanting to take a vacation to the Caribbean next year. I know there are different islands for different tastes, and I want to have the best experience for us. So, here's what's important/not important to us:
IMPORTANT: great beaches with clear water and accomodations on or very close to the beach, NO small children, low crime, secluded/less crowded, beachside bars, low season rates starting in April/early May.
NOT IMPORTANT: shopping, nightlife, 5 star restaurants, watersports.
Also, I don't want to have to rent a car. I'd like everything we need to be around the hotel, but I'm not a fan of the all inclusive places.

From my limited research, I'm thinking the BVIs or St. John. Also thought about Bahamas or Turks and Caicos. Thoughts? Any specific resorts/hotels to recommend?

both st. john and turks and caicos would require a car or lots of taxi rides, but should offer you the secluded/less crowded atmosphere.
bvi and st. john are both lush hilly places. t&c and bahamas are flat, scrubby islands.
if you truly want no kids you are greatly limited in your resort choices.
you might check out barbados. if you stay at a resort near 'the gap' i believe you can walk to things. but this is not a secluded kind of area.
to offer any real advise on specific resorts you will have to give us a budget range. also, where are you flying from and do you care how long the journey is. st. john for example requires that you fly to st. thomas. taxi to ferry. ferry to st. john. taxi to resort or villa. bvi's is fly to tortola, then fly or boat to other islands.
by traveling before school gets out i'm thinking you can avoid most families with children no matter where you decided to go...

The taxi thing is not as much an issue as the others. I wouldn't mind the occasional cab ride, I just don't want to have to do it everyday.
We are on a budget, and I've heard Barbados is great but pricey. Probably need to keep accomodations under 200/night. Flying from Atlanta or Orlando. Transferring in Puerto Rico/USVI is not a problem.
What about Jost Van Dyke? I know its really small, but it seems like the kind of place we could just sit on the beach all week and relax, and maybe take a day trip to one of the other BVIs.

Actually Turks and Caicos fits pretty much all your criteria. The hotels are lined up (not close to one another) on Grace Bay. As long as you do not stay at Sands/Ocean Club/Beaches you will not find kids (or very few). I would try to find a central location so that you can access some of the other restaurants/beach bars by walking on the beach. You could also take a taxi to some of the other restaurants if you want access to more selection

Jost Van Dyke is very small. You may want to look into Tortola as well. I believe there are some reasonably priced on beach resorts there. Also, you may wish to look into renting a small cottage or condo. Low season does start in April/May, and the price for a rental even on St. John can be as low as $1000 a week for a couple. Having your own refrigerator and kitchen also cuts down on food/drink costs.

I agree with Turks & Caicos. Grace Bay is fabulous. We stayed at the Sibonne. It's a small boutique hotel with a resturant and bar. Only 26 rooms and does not cater or have amenities for children. We went in May' 06 and there were no children at the hotel for our stay. We also got an upgrade to a room with a seperate living area. Very nice. The rooms are literally steps from the beach, especially is you get the beach apartment.

The Sibonne is within walking distance to the Ports of Call or Salt Mills for restuarants/bars. We walked almost every night and had no problems feeling unsafe. Didn't see any beach bars on Grace Bay, but the Sands is next door and the resturant/bar was pretty close to the beach. We saw lots of kids there though. Taxis were around $6 to Salt Mills/Ports of Call. $12 to Turtle Cove. I think we paid $16 to the Geico Grill. (those are one way prices) The taxis were more convenient than the Geico Bus. We did not rent a car and it wasn't an real issue for dining, but we wished we had rented a car for at least a day or two to drive around the Island.

BVI - Tortola, best inexpensive place to stay with great food is Heritage Inn, set in Carrot Bay, innkeeper on site, great bar and best sunset views. Food is inexpensive and terrific, Christine the innkeeper is friendly and you can actually use the full bar on "honor" basis when left unattended. As well, short cab ride to Cane Garden Bay, terrific beaches, restaurants and of course the Bomba Shack.