Bring a small pot of water to a boil and have a bowl of ice water nearby. Add the basil, stems, and all, and blanch until the basil has wilted but retained a bright green color. Immediately transfer to the bowl of ice and cool. Once the basil is cool, remove it from the water, squeeze as much water from the basil as you can, and basil it in a small food processor or blender jar. 

Add the mayo, salt, and garlic. Puree until it is as smooth as can go. Transfer to an airtight container and store in the refrigerator for up to a week.