Sorry I'm a bit late on this topic, I did a very similar job about 5 years ago and its still going strong.
The method I used was to fit two parallel 12mm x 12mm beadings underneath the handrail and along the stair string (open side of course, but it would work on each tread as well depending on the stairs construction). The gap between the beadings was the width of the balusters.
I then got a length of 12mm thick timber (same width as the balusters) and used this to infill between each baluster (basically constructing a
12mm deep tenon that the baluster could be slotted into). It has the obvious added benefit of removing any concern about setting up the gaps as each infill piece does this job for you.
In terms of the gaps between I made the infill pieces 90mm on the horizontal run and about 120mm on the angled runs (with each end cut at the appropriate angle, of course). I got two small children and although I get a wide variety of toys raining down on me at times, no children to date.
Cheers
Richard