Holden Lipton spent last Christmas indulging in mini-mugs of hot chocolate with marshmallows by the Jacuzzi at the St. Regis Resort in Aspen. He’d spent Thanksgiving in a much warmer, but equally luxurious, locale: The Four Seasons Resort in Punta Mita, Mexico, where he registered at a dedicated check-in desk and enjoyed some of his favorite dishes at gourmet buffet lunches.

Holden Lipton is four years old, and he, like his peers, is visiting sophisticated destinations with his parents. A recent survey by American Express’s Platinum division found that over 90 percent of its affluent cardholders travel with their children. This new breed of worldly traveler—aged 15 and younger—is accustomed to fine food and designer labels, just like their parents. For them (and their parents), a typical kids’ club and plate of chicken fingers just won’t do, and so properties are offering activities, amenities, and menus aimed at satisfying refined tastes.

“I want to be with my children on vacation,” says Holden’s mother, Tara Lipton, co-founder of Citibabes, a chic members-only children’s play space in New York. “And I look for a place where my children can be part of the activities, but I don’t want to be uncomfortable.”

Sheilah Benoit, a family-travel specialist for American Express, has seen requests for destinations that can accommodate savvy kids and their families double in the last 12 months. Particularly popular are resorts where children can learn about local culture (with language lessons, for example), instead of just playing on the beach. “More and more, there seems to be requests for a program that will attract the children’s interests,” she says. “Properties know they can cater to what parents want and also offer children something they’ll be pleased with.”

The Fitzpatrick Hotel, for example, offers an “American Girl Experience” package at its two Manhattan locations. Each child’s doll is treated like a V.I.P. guest, with its own miniature bed, tiny slippers, and turndown service each evening. The hotel staff is specially trained to speak to the dolls, who register alongside their owner and her parents. Breakfast, with dolls in high chairs at many tables, may seem surreal to some adult guests, but the package has been a smart business move. Since its launch three years ago, around 5,000 families have chosen it.

In May, the W New York launched “Wee New York,” its first child-specific promotion, which includes customized treats from Dylan’s Candy Bar, a $25 gift card for toy mecca F.A.O. Schwartz, and discount cards for a handful of nearby stores and restaurants. Guests who need restaurant reservations are pointed toward chic but child-accommodating hot spots like Da Silvano and Balthazar. The hotel anticipates the package will attract 200 families by the end of the year, with an uptick in demand to follow in coming years. (Prices begin at $239 per night.)

The trend is not limited to urban destinations. The Grace Bay Club in Turks and Caicos dubs its youngest guests V.I.K.’s—very important kids—and offers them their own mini culinary academy on the premises. The recently opened Cove at the Atlantis in Paradise Island features specially designed workout classes for younger children. The JW Marriott Cancun Resort introduced an extensive range of children’s treatments in its spa two years ago.