If "coarse" refers to texture:
* Fine
* Smooth
* Delicate
* Soft
* Refined
If "coarse" refers to behavior or language:
* Polite
* Refined
* Elegant
* Gentle
* Tactful
* Sophisticated
If "coarse" refers to quality or grade:
* High-quality
* Superior
* Premium
* Fine
If "coarse" refers to something being rough or unrefined:
* Subtle
* Delicate
* Sensitive
* Refined
* Elegant
The best antonym depends on the specific meaning of "coarse" in the context you're using it in.