[untilcomplete]

I've searched for "stair rail base" and the stair section of the hardware store. No luck. Need to replaced a small section that split.

 

[WhatRnsdownhill]

I think its called a shoe rail...but yours looks custom made from oak..

 

[mako1]

http://me360e.com/stair-rail-components.html

 

[carpdad]

I don't think there is a name for it, unless used by carpenters in california or by an architect.:smile: The balusters are almost always on the stringers or on the treads.
It is just a finish grade lumber. Some split wood can be repaired. If you have the work room, a butterfly joint is one decorative repair. The balusters can be reinforced with longer nails that bite into the frame. Predrill first.