Reach the upper Runswick Bank Top car park from the A174 on the signed Runswick turn. There is a smaller car park down near the beach, but it is more constrained and can fill, so use the upper one and walk down through the village. Dogs are welcome on the public route and should be kept under close control near livestock, birds, other walkers and the unfenced cliff edge. Short leads are advised around livestock, and seasonal lead control applies where ground-nesting birds are present.
If you want something meaningfully easier and more accessible on this stretch of coast, the short Miles without Stiles route at Runswick Bay is the better pick.