Christopher Till and Paula Radley pleaded not guilty at Tain Sheriff Court. They were released on bail with special conditions including an order that they leave Scotland and not attempt to enter the waters of the Cromarty Firth.
Greenpeace began an occupation of the rig on Sunday. It is protesting against the drilling of new oil wells.
Mr Till and Ms Radley’s bail conditions also include an order that they only re-enter Scotland for pre-arranged appointments with their legal representative and court hearings.
However, they will be allowed in Scotland for pre-arranged holidays as long as they do not go near the Cromarty Firth.
They have been accused of conducting themselves, while acting with others, in a disorderly manner and boarding an oil platform without permission or other lawful authority on 9, 10 and 11 June.
The charge also accuses them of refusing to leave, attaching themselves to the platform by tethers, placing themselves and others in danger and preventing other persons going about their business and committing a breach of the peace.
The Transocean rig, under contract to BP, was due to leave the Cromarty Firth near Invergordon on Sunday, heading for the Vorlich oil field east of Aberdeen.
But the operation was halted after two Greenpeace campaigners boarded the structure on Sunday evening.
Greenpeace said two protesters were still on the rig and that they had sufficient provisions to remain there for days.