Do I need to be 'current' or hold a medical to become a Basic Ground Instructor?
You do not need a medical certificate.  It also does not matter that you have not flown in a long time.  You will need to study and take two separate tests to get you ground instructor certificate.

Fundamentals of Instruction (FOI) Test
Ground Instructor Test

The FOI information is covered extensively in the Aviation Instructor Handbook and is 50 questions (at least it once was).
The Flight Instructor test question bank is also used for the BGI (Basic Ground Instructor) and AGI (Advanced Ground Instructor) tests.  If I remember correctly, the BGI test is 80 questions and the AGI test is 100 questions.  The information on the test is covered in multiple FAA handbooks (not all inclusive):

Pilots Handbook of Aeronautical Knowledge
Airplane Flying Handbook
Rotorcraft Flying Handbook
Balloon Flying Handbook
Aviation Weather Services

One thing to notice is that the ground instructor test covers all aircraft and not just the &quot;airplane category&quot; as you can teach and sign off written tests for all aircraft.  Since the BGI and the AGI are the same test, I would suggest getting an AGI as you will be able to sign off all written tests and not just tests for sport, recreational or private pilots.