Resorts below include several all inclusive options and one legendary luxury resort that welcomes families.
Almond Village Resort
An all-inclusive resort on the west "Gold coast", very popular with visitors from the UK.
Almond Casuarina Beach Resort: another all-inclusive by Almond Resorts, this one on the white-sand Dover Beach on the south coast of Barbados.
Bougainvillea Beach Resort
On the "cheap and cheerful" south coast, this mid-sized resort has 100 spacious suites on 2.5 acres with free-form pools, swim-up bar, and waterfalls. Meal plans are available. The Bougainvillea is "Green Globe" certified, meaning it meets certain environmental standards.
The Fairmont Glitter Bay
The Caribbean With Kids guidebook says "you'll feel the elegance and refinment of Barbados and Britain" at this resort, built as the home of shipping magnate Sir Edward Cunard. Kids are welcome, with kiddie pool and kids' club.
Sandy Lane Hotel
A top-end resort, for those seeking luxury and prestige. Kids are welcome, with Tree House Club for ages 3 to 12, including some evening hours; Toddler Corner and The Den Teen Center.
Turtle Beach Resort
Part of the Elegant Hotels Group on Barbados, Turtle Beach is a "posh" place on the South Coast, with 1500 feet of white sand beach; and Kids Club. (Several other Elegant Hotels are on the upscale West Coast; some have kids' clubs.)
Beyond resorts: many families visiting Barbados rent villas/condo's; the busy South Coast is a prime area. See Barbados Accommodation.
--Please note that the writer has not visited all these properties in person; and always check resort sites for updates.