How can I secure my spindles?
Stair hand rails are primarily supported at the top and bottom by the newel posts. While some additional support is needed in a span over 5 or 6 feet, the handrail does not need dozens of braces. You will often see an inside wall handrail with attachments that are several feet apart. The spindles are primarily intended for safety, to prevent someone from slipping under the handrail on an open staircase.

No spindles on a full handrail run would allow the rail to flex too much, so they are necessary. But if a number of them are loose, there is little likelihood of risk unless the attachment at the newels is also compromised.

Having said that, it makes sense to firm up the spindles. Driving 1 1/2 or 2" finish nails on an angle, pinning the tops and bottoms of the spindles to the handrail and the base should probably do it (much easier with a nailgun).