There has been no train to the village for many years but there is a regular bus service to Harrogate which operates seven days per week. This is service 24.
On summer Sundays and Bank holidays it is possible to get a bus from York (812), Leeds (820) and another Harrogate service via Brimham Rocks (825).
The linked website has full details.
If you don't have your own car, it can be awkward to get here but don't think it is impossible. Pateley Bridge is on the way to many other places in the Dales and so, buses go here from for instance Harrogate which is easy to get to.