Hi Babyjojo,
We've just taken our nearly 2-year old to Greece - have posted about it last week as follows. Hope it's helpful. It's a really good value villa option. Our place was very clean and well maintained too and the gardens are stunning:
...We've just come back from a two-week break staying at this place and would really recommend it to anyone looking for somewhere baby/child-friendly, low-key, walking distance from two beaches, very quiet etc. It's run by Maria and her husband Dionysis who speak an impressive array of languages and were so helpful and welcoming. It's in Porto Roma on the south peninsula about 20 minutes walk from Gerakas, one of the protected beaches where the loggerheads turtles nest, a beautiful bay with shallow waters perfect for kids, and a safe distance from 24-hour Laganas (Brits abroad...say no more). It's set in a beautiful garden full of flowers and birds and an olive grove. Each villa/apartment is individual and built in the traditional Greek style with wide veranda/wooden shutters.
There is quite a lot to see on the island - we hired a car for three days. You could get away with not hiring a car - Gerakas is a beautiful beach - and we appreciated the option of not having to commit to hiring one when we booked the holiday (Daphnes picked us up from the airport).
There is only one two-storey villa so you woudln't need a stairgate - Maria gave us the use of a highchair and travel cot and baby bath etc. There's a small children's play area - no pool but it makes the place a lot quieter.
If you baulk at the idea of being organised by a rep and like to travel a bit more independently (flights go from Gatwick direct to the island), this is a really good alternative to the Periyali Villas type of holiday (which incidentally is round the corner from Daphnes). Greece may not be cheap any longer but Daphnes was lovely and very good value.
They are at www.daphnes-zakynthos.com