I don't think it will be a huge problem. People aren't going to commit more crimes so the caseload there won't increase much. "Jailhouse lawyers" might file more appeals of their own convictions but those seldom go very far.

On the civil side, high-dollar disputes are already litigated. The real potential growth enabled by cheaper legal services will be in small claims. Those courts can be scaled up because they don't require a jury or even a real judge.