Two places that meet all your criteria are Curacao or [-]Grand Cayman.[/-]
edited: Cayman isn't a "Lesser" Antille.
Curacao will have on-site dive operations at many resorts - even some of the condo complexes have their own. A couple resort options to consider there are the Marriott Emerald Beach in Piscadera Bay or Lions Dive south of town. Both have on-site operators, beaches, shore dives, etc. For the kids Lions Dive is also walking distance to the Sea Aquarium. There's also a row of boardwalk shops they recently developed -it's all cobblestone paths. There's a few other restaurants in that area also. Lions Dive has a 100m pool plus a really shallow kids pool next to it. Some of their bigger villas have full kitchens also. Ocean Frontiers main location is 50 yards to the south - Curacao's largest dive operation. There's also a boardwalk that runs to it and past it - IDK how far. I suspect it goes all the way to the Sea Aqarium which is located on a small island just across the channel. If you're interested in Dolphin dives - they do one with the S/A's dolphins once a week. Not the best shore dive there but it's decent along the breakwater. Saw the only pipefish we've ever seen on Curacao there.
If you want to leave the car with the folks, the Dive Bus is an option. They do escorted shore dives all over the island and pick up anywhere in town. Bas Harts is another guide option. As is Atlantis Diving/Apt's south of town. I'd recommend them but their beach is all ironshore/rock.
The Marriott in PBay is also very nice - probably one of the nicest Marriotts I know - and I'm a rewards member. Nice beach, pool complex, Caribbean Sea Sports on-site for diving. Don't confuse it with the Renaissance which is downtown, adjacent to the cruise port and has the large casino. Since downtown is on short cliffs, there's no beach. Somehow they have a sandy beach around their pool instead.
PBay is a small convention area but quiet - at night really quiet. Small casino there and at the Hilton next door (which btw is charitably a dump) There's also other restaurants within walking distance but more hard paths along the side of the road than real sidewalks.
Curacao Official Site - A Unique Island Paradise is a start.
One other advantage for Curacao is that many of the western beaches are seriously nice - wide stretches of white sand often with operator improvements, thatched huts, tables etc. Often they request a small fee to use the facilities. Bathrooms also and most have at least a snack shop - at Varsenbaii there's a decent restaurant. Just about everyone is also a shore dive site with an operator on site for tanks - often the same people run the whole concession.
Some parts of Curacao can be a little gritty - there's both a deepwater container port and an oil refinery near downtown. And some theft - most resorts will be gated/patrolled. We stayed in some really quiet apts. outside of town for a few days - bars on all the windows/alarm system. Not for us but to keep locals from "browsing" when no one was there. The mgr. also told us to hide/lock stuff in the vehicles at most of the beach areas. Some of the operators will watch your vehicle if asked - usually the shops are within/behind the parking areas. A couple like Porto Mari - a popular locals beach - have a security guard. After we moved back into town, our duplex condo complex was fenced, had a guarded entrance and a watchman walking around all night.
Any of the populated areas seemed nice and safe though. The Dutch military made themselves known a few years ago when Curacao went back under Dutch protected status - the whole island was in debt till the dutch bought them out. My friends wife is a native so they go back to see her mom almost yearly - she noticed a big difference. In case it's needed for the kids, the hospital there is named the nicest one in the Lesser Antilles.
When they read "shore diving" someone will suggest Bonaire also. Which is a good option except that to dive anywhere but your resort requires a vehicle. And no dive site outside of those in front of the resorts have any sort of facilities - none. No bathrooms, shade etc. I can name 3 resorts with some semblance of a beach, Harbor Village, the Divi Flamingo or the Plaza Resort. Sand Dollar Condos has a small one also if you're considering it. None of them are great since there's ironshore along most of the Plaza's and reef in the water at the Divi. Unless they've changed that now, they did a major reno this year. You'll notice most condo rentals on Bonaire say Oceanfront - not Beachfront. That's because most are - ironshore/rubble to the water line. Some have stairs to the water also. Some of the resorts - like the uber-popular Buddy Dive - have sand dropped on top of the ironshore. Still requires a ladder to access the water though.
