An open tread staircase such as this may have been trendy at one time but nowadays it’s an accident waiting to happen- especially for small children, the elderly and even your beloved pets!
Think about enclosing the spaces between the “treaders” and the “risers” with timber pieces which can then either be painted/varnished or carpeted over.
This is a simpler and cheaper option to renewing the staircase completely although replacement is the safer and preferred option.