By 11 a.m. on a recent sweltering Sunday, Harrison Anastasio had already collected the cellphone numbers of five bikini-clad women on the rooftop pool at the James New York Hotel.

"Don't forget to text me," one of the women whispered to the 17-year-old, flashing him a coy smile. Anastasio colored slightly beneath his Wayfarer sunglasses. But he wouldn't forget. Though it sounds like the setup for a 1980s sex comedy, it's his job.

Anastasio,...