How to make a 19' bannister sturdy?
The spindles provide virtually no side-to-side strength; they're all to carry vertical loads. So, improving their attachment won't help at all.

You have two choices:


Install a stronger rail that won't flex as much. This could be a thicker rail, or perhaps an additional strengthening piece added to the bottom.
As @bib suggested, support the rail in the center with a second newel post, well-anchored at the base.


Of course, it's been this way since it was built, and sounds like it's a safety hazard. Talk to the builder?