You can choose from the government-run hospitals, such as Al Wasl, which are cheaper than the private hospitals, but many do not allow men into the labour rooms (they are allowed in for final delivery only). Al Wasl has an excellent reputation in terms of quality of healthcare, but is far from providing any frills and luxury.
The best and most established private hospitals are Americal Hospital and Welcare Hospital. Be aware that American has a tendency to push women into Caesarians although it provides a very luxurious service complete with champagne dinner for the new parents.
Welcare is not quite as luxurious but the care is very good; mainly British/Indian doctors and nurses.
There are also a handful of new hospitals cropping up such as MedCare in Jumeirah, which have yet to establish a track record, but have super facilities.