What happens to the current when a wire is left open to air at one end?

Since the final state might well be that no charges are moving you could treat it as a question about electrostatics.

You have an uncharged wire and as you bring it close to the negative terminal of a battery (surplus of electrons here) then charges are induced on the wire by the free electrons in the wire mooing away from the negative terminal leaving a surplus of electrons at the other end of the wire.
This movement of the free electrons continues until the electric field inside the wire is zero and one end of the wire is connected to the negative terminal of the battery.

The air around the wire could become a conductor if the air is ionised accompanied by the production of mobile electrons.
The ionisation of the air can be achieved by having a radioactive source close by, by heating the air or by having a fairly large electric field around the end of the wire which then ionise the air via an avalanche process.

The free electrons in solid and liquid metals are there because it is energetically favourable for a lattice of positive ions to form with the liberated electrons being able to wander throughout the metal.
This is the essence of a metallic bond.

However the electrons are confined within the metal as it requires energy to remove them electron from the solid.
The photoelectric effect, thermionic emission and field emission are all examples of electrons being given enough energy to escape from the metal.