US residential stair railing code: What is the requirement for terminating an open railing?
That handrail does not meet code, because it is not continuous from top to bottom of stairway and the ends do not terminate correctly at the bottom. 

The Code (ICC R311.7.7) requires handrails: 

1) Height to be between 30” and 38” above the nosing of the tread, and

2) Be continuous on at least one side of a stairway with 4 or more risers and be from a point directly above the top tread to a point directly above the lowest tread. (It can be interrupted at turns,) and

3) The size is to be between 1 1/4” and 2” if it’s round. If it’s not round, then it needs to be 4” or more around but not more than 6”. If it’s bigger than that, it needs finger grips. (It gets complicated because the finger grips are restricted too.)

4) If it’s located next to a wall it shall have a minimum of 1 1/2” clear space to the wall, and 

5) The handrail end must terminate to the wall, newel post, or safety terminal. Safety terminal is not defined in the code. 

The intention of returning handrails is to keep it from catching someone’s pocket, etc. Btw, we design a lot of handrails and when only one is installed, we prefer it on the right side descending. 

The code does not require tube handrails to be capped, as Ed indicated.