They still have relevance. Lets take the Antartica treaty for example. Lets, for the sake of argument, say only one entity has the knowledge, resources, etc to reach Antartica and this entity is a subsidiary of BP. BP-Antartica lets say. BP-Antartica wants to drill for oil. This would still be a violation of international law under the treaty that governs Antartica. Same thing applies, or should apply to SpaceX or any other entity (public or private) with regard to any existing and future space treaties.

"International law identifies Antarctica and outer space as two of four global commons (the others being the high seas and the atmosphere). Both have been defined within international regimes as spaces outside the territory of nation states and beyond the normative inhabitable zones of the human species. In both instances the current governance regime were outcomes of the International Geophysical Year (IGY) (1957-1958)."

http://phys.org/news/2015-08-antarctica-key-space.html

Specifically in the Antartica treaty:

"No acts or activities taking place while the present Treaty is in force shall constitute a basis for asserting, supporting or denying a claim to territorial sovereignty in Antarctica or create any rights of sovereignty in Antarctica. No new claim, or enlargement of an existing claim to territorial sovereignty in Antarctica shall be asserted while the present Treaty is in force."

https://en.wikipedia.org/wiki/Antarctic_Treaty_System