Flip over part of the curtain so we can see both the face and backing of the curtains in the same photo. Clear and close-up. The best procedures will depend upon how they are made and any backing present.
Todd's advice to use a professional rug washer is excellent.
If you want to try it yourself, mix Bridgepoint's Coffee Stain Remover (other coffee stain removers may not have the same formula) about half the strength called for on the label. Use warm water. Apply to small area with a little paint brush. Wait a few minutes to see reaction. Do this work in sunlight will increase the action.
Increase the water temperature and strength in small steps until you get something that works. Be careful of getting it too strong. It can leave the fabric whiter than it already is. It can also weaken the fabric making it easier to rip or tear.
Looks like a Jacquard type weave. Some of the red from the back has bled through to the front.
Rug specialist is still a good idea.
For doing yourself, I would follow what the stated above but add vacuum to the back side of the fabric at the point you are working on. The small Water Claw Spotter might be a good tool to accomplish that.