We were allowed on the Bellagio property (checking out the fountains), in March '02 with a 10 & 13 yr old. Unless something has changed, no one seemd to care that we had kids.

It is usually inside the Bellagio. I would hope they wouldn't stalk the fountains.

Not that this really matters, but we were viewing the fountains from the inside. We were only there for a very short while, literally just passing thru and catching the 'dancing' water. They may not want people lingering in the public areas with kids, or certainly not kids running around (i don't think any hotel would want this).