Roosevelt Hospital in Manhattan offers private post-partum rooms for $750 a night, The New York Daily News reported. The fee includes bed space for the baby's father or a "support person" and concierge service as well as hotel-style amenities like toilet items.

"The occupancy rate of the new wing has been close to 100 percent," said Timothy Day, an administrator at St. Luke's-Roosevelt. "There is a real demand for more upscale private post-delivery rooms and we have been quick to respond."

Mount Sinai has a sliding scale with private rooms going for $500 to $850, depending partly on the view.

Kira Krieger, managing editor of The Bump, a magazine providing information for expectant moms, said the growing number of older parents has contributed to the trend. She said they tend to have more discretionary income.