As far as I am aware, we would be using a what is basically a floating platform which could be moved as necessary. The nearest equivilant currently would be more of a FPSO, which is a modified supertaker which can produce and hold oil taken from seperate drill rigs, refine it onsite then discharge it to smaller shuttle tankers.
These still have to be registered to a nation though and still are covered by the normal rules and reg's that govern the sea (MARPOL, SOLAS, IMO, flag state controls, registry controls, etc, etc)
I can't see why this would be any different for the elevator platform. You could register the platfrom in whatever country you want. For ease it might be better to flag out to a country such as Liberia where the rules are less rigid that say the US.

The Space Elevator will need laws of its own.  Where ever possible these should be similar to those of ships and buildings.  In may ways it counts as:[list=1]

• A ship.
• A very tall tower.
• A road/railroad - to the lifters.
• An international airport - to people landing on the ship.
• A space landing/attachment port at Geosynchronous Earth Orbit GEO.
• A satellite launch point along practically the whole of the cable.
• An obstacle in space.

Civilian ships are not authorised to act as air traffic controllers, without such legal authorisation the Space Elevator company cannot have misbehaving pilots arrested.

The ribbon is weight limited so "landings" in space, sea level attachments, departures, launches off the ribbon and movements along the ribbon have to be strictly controlled.  Breaches of these rules can result in crashes and cutting of the ribbon.  Such incidents are likely to result in deaths, injuries and destruction of property.

Without traffic control along the ribbon lifters can crash into stationary lifters, slower moving lifters and lifters going in the opposite direction.  Using contract law and trespass law without criminal sanctions to enforce the rules may not be easy.

Transport companies like railways are normally given "common carrier" status with the corresponding powers, duties and responsibilities.  The Space Elevator's licence will probably need to supply this.  Moving vehicles and rail tracks are subject to inspection - who employes the inspectors (UN/USA/Lloyds Insurance?) and what powers should they have?

We may not need a trafic law until say the third Space Elevator but it is easier to pass one 10 page law than say five 2 page laws.  There is plenty of time to get the law right.