All our One Bedroom Apartments have balconies - so I recommend you either book a City or a Harbour Apartment and there you go! (None of our Hotel Rooms have balconies)
Ask for a room looking out to the Harbour. These have a balcony but they are small.
We booked a self contained unit. It had a small kitchenette, laundry in the bathroom, lounge room and a small balcony off the lounge room. It did not have a fantastic view being at the side of the hotel, but you could see the bay and it was pleasant to sit out there.
You can request a balcony and they will try there best to get that for you but all 1- 3 Bdrm Apartments have balcony's. Try to get the Harbour side you may pay a bit more but the sunset and the view is just fantastic great way to enjoy the beautiful scenery and weather Darwin has to offer. Nice hotel, great spot close to Mitchell street pubs, cafes, shops etc.
They still will not guarantee a balcony despite having done this!
Get an answer from Mantra on the Esplanade staff and past guests.