Does the line ever move?
If not, then it is likely a hardware issue. Bad cable, bent pin, faulty panel, some other internal monitor issue. If you've already tried different cables and the problem is still there but is not moving, always in the same place, then it's doubtful there is anything you could do to fix it. Replacement of something internal or the whole unit would be your options.
If it moves or changes, it could be a driver issue or something with the graphics card, maybe. I'd bet on it being 99.9% a failing monitor.