Are spindles strong enough on a balcony to keep a toddler safe?
Spindles (also called balusters) on a staircase are supposed to be properly spaced, structured and attached to prevent a toddler (or anyone else) from falling under the railing. 

Typical spacing is 4" on centers, too narrow to fit even a small head between, but check your local code. Obviously children could insert a limb and get hurt, but collapse is unlikely.

Standard balusters are usually made from hardwood, but even softwood versions should be sufficiently strong to handle the side pressure load of a leaning or falling person on the stairs. Any load is likely to be spread over several balusters.

Balusters are attached on the bottom either by being inset into holes in the tread or toenailed to the tread.  They are also toenailed into the handrail. This is generally sufficient to handle most strains including the pressure of a forty pound child (and probably a 140+ pound adult). Again any load is likely to be spread over several balusters.

If your handrail and balusters are substandard or improperly installed, you should repair the stairs.

If you just want to block the gaps to minimize limb insertion, any thin, smooth coverage will do. Pre-drill and screw into the balusters.

You also may want to think about not letting toddlers use the stairs without supervision.