remember that this is an electrostatics problem, the current will decay exponentially with time (sorta). I'd image more realistically that initially upon dielectric breakdown of the air the current will rapidly increase due to the decrease in resistance as the air is ionized, however this action will be counterbalanced by the fact that the charge on the finger and therefore the electric field across the air gap will be decreasing rapidly. At some point after the current has risen to a peak value the second action will begin to dominate and the current will begin to go down, as this happens the energy going into ionizing the air is also reduced meaning the resistance will increase again further limiting current. As a result the ionized particles will recombine much faster than they are formed and eventually the dielectric strength of the air will return to a high enough point such that spark will be completely extinguished.
In summary, as you suspected the current is clearly very small and further the time duration is very small so the total energy released is not enough to cause any harm (although it sure can hurt sometimes).