I’ve been asked to clarify if the tag can be applies to nodes, ways or both.
This is a tricky one because there are different mapping styles for crossings. Some mappers put crossing tags (such as crossing:markings) on the highway=crossing node and the way only gets highway=footway, footway=crossing and typical way tags (e.g. surface) (example). Other mappers duplicate the crossing tags on the way. Yet other mappers do not tag the node at all (example way with untagged node).
Therefore I suggest the following:
The tag can be used on nodes (with
highway=crossing) or on ways (withfootway=crossing), but there should be no need to use it on both for a single crossing. When ahighway=crossingnode is present, it is considered sufficient to just tag the node withcrossing:continuous=*.