There are heaps of villas at all price points, depending on which area of Bali you're wanting to stay in.
I went with my then-18mth son and my parents to Seminyak in October 2011.
We struggled to find villas where the pool is not directly off the living area - everything is open-air Bali-style pavilions (or fishponds next to walkways etc). If this doesn't bother you, it will be easy to find heaps of places just by googling. It is also possible to hire a pool fence or get your villa to arrange this for you.
After much searching we found something (in our low-mid price range) where the living area had glass doors all around, so we felt much safer being able to close them as needed. http://www.homeaway.com/vacation-rental/p453908
I have also previously stayed in Sanur, with my parents, and sister plus 11mth old nephew. He was just cruising, not walking so was easier to contain/stay near. We stayed in a lovely 3br villa in the open-air style: http://www.baliemeraldvillas.com/
Happy to answer any questions you might have
You could also try posting in the Bali Travel section/topic - it is an ongoing thread with lots of information to absorb, but I think most posters in there stay in hotels (?).