For the action of opening or closing a zipper:
* Zip: This is the most common and informal verb.
* Unzip: This is the opposite of "zip."
* Fasten: This is a more formal way to say "zip."
* Unfasten: This is the opposite of "fasten."
* Close: This is a general term for securing a zipper.
* Open: This is the opposite of "close."
For the action of putting a zipper into a garment:
* Insert: This is a general term for putting something in place.
* Install: This is a more formal way to say "insert."
For the action of fixing a zipper:
* Repair: This is a general term for fixing something broken.
* Mend: This is a more specific term for repairing a zipper.
The best verb to use depends on the context of the sentence and the level of formality desired.