There's a B&B on Grand Cayman.
Bed & Breakfast Inn Grand Cayman Around $700/wk which is cheap for Cayman. I've never stayed there as we rented condos on SMB.
On Curacao we stayed at Piscadera Bay Resort, it's showing as $92/nt. Which seems about right, for the week with the cleaning and electricity my bill was around $650. Last April.
Piscadera Bay Resort :: Home (eng)
Every condo is a duplex and has two bedrooms. I'd guess the age of the units as mid-70's. All only have A/C in the bedrooms, mine only had wood shutters on the windows. My buddies had been refurbished recently - ask for #4...Clean, just not the newest. Since they're in PB they have gated security with night patrols. Although PB is pretty quiet at night.
Just inland from the Marriott and the Hilton so you have dining/diving/casino options. Also check out Hooks Hut just down the street it's a diveshop/restaurant/beach/bar combo. Good for lunch. They also rent tanks for shorediving off-site.
For something farther west (better diving) we stayed in clean, newer apartments in Lagun.
My Dream Apartments
We were the only people there at the time so it was pretty quiet. About a 50-60' vertical ladder down to the water there so you'd want to drive to Playa Lagun or Playa Jeremi to dive. Reasonably close to West Punt also, maybe 10-15mins.
They said there were some security issues judging by the 8' walls and bars on every door/window. But no issues the 3 nights we were there. The owners Max and Sonia dropped by once or twice just to see how we were doing - nice people.
All the units were 1BR with a typical Curacao kitchen and small living room. The oceanside units have a great view/breeze. I don't remember what I paid but it was really reasonable, as I recall maybe $250 for 3nts?.
Definitely need a car out there and stock up on food in Willemsted before you drive out. At Centrum market, just a couple miles south of the airport. They even sell cheap styro coolers for the drive out.