Incorrect argument for command at position x
With the new execute syntax, you must put run before the command you want to run, like this:
execute at @a[x=0,y=4,z=-0,dx=15,dy=1,dz=0] run worldborder center 8.0 -8.0
The error is saying that worldborder is not a valid argument for execute, as it's expecting an execute subcommand (as, at, etc.)