Does traditional Pesto Genovese contain garlic?
According to the CONSORZIO DEL PESTO GENOVESE, which defines what is considered official Pesto Genovese, it does contain garlic. They suggest that traditionally it contained less garlic than the current official recipe calls for - one clove for 600g of pasta versus two.  Later on they mention that it contains one clove for each thirty leaves of basil, and also leave the following suggestion:


  The garlic must be sweet, it must not prevail while making itself felt in the background ... in short, it can not be missing!


Note that pesto is traditionally not cooked at all; it’s made solely in the food processor (or of course a mortar and pestle or mezzaluna truly traditionally).  

As far as your taste - I would suggest leaving out the garlic if you truly don’t like it!  

Some suggestions for keeping the garlic taste down:

From Cooks Illustrated:


Blanching the garlic (briefly cooking it in boiling water).  
Microwaving the garlic until warm
Toasting the garlic in a dry pan


From Fine Cooking:


Remove the germ, or the thin center piece of the garlic, which often turns green (like a sprout); this is more bitter than the rest of the garlic.


Avoid cooking it in butter or other preparation methods that will substantially alter its flavor, as that will also alter the flavor of the pesto excessively (unless you like that flavor change!).