2003 Giant Rainier MTB Leaking Shock Fluid?
The spec fork on that was the Answer Manitou Axel Super -  information here   

From the 2003 Axel user manual.... 


  CHECKING OIL LEVEL IMPORTANT: Setting the proper oil level in your
  Fluid Flow damped fork is critical. Fluid Flow damping is located in
  the right leg of your Manitou AXEL fork. Not enough oil will allow
  foaming and reduce the performance. Too much oil will restrict travel
  and may cause damage to the system and create an unsafe riding
  condition. Finish reading this entire section prior to altering the
  oil level. To check the oil level on the Manitou AXEL, remove the
  compression assembly located in the top of the right leg (as you are
  looking at the fork from the rider’s position). Leave the left side
  spring stack in place to keep the fork fully extended. Use a tape
  measure or “dip stick” to measure from the top of the fork crown down
  to where the oil sits (Figure 2). The oil level for the Manitou AXEL
  should be per following table. Fork Model Oil Level Manitou AXEL 
  115-125 mm (4.5-4.9”) NOTE: Use SAE 5WT fork oil suspension fluid
  only.
  
  Forks need regular servicing - anything as often as every 25 hours
  with seal replacement commonly being recommended at 100hours riding (I
  think Manitou had relatibvely long service intervals . Seal
  replacement and lower leg service is not too hard, but is a bigger job
  than many owners are prepared to tackle themselves.  A shop secive
  with seal replacement will be around $US100 (various with you location
  in the world).


The seals have gone - possible just from old age and wear. Given the age and value of the bike, its likely not worth the cost of new seals, especially if you are getting a shop to do it.  If you don;t want to pull the fork apart, replacing the oil is straight forward, but I would be tempted to do a lower leg service and then top up the oil regularly.

The oil leaking is a messy problem, but no causing functional issues as long as there is oil to leak out. If the leak stops its time to worry.