The problem with American basil is that it is too strong. This becomes even more acute in summer when locally grown basil is available. To mitigate this, use only the small leaves from the plant, or grow your own, preferably from seeds labeled basilico Genovese. Most references to Italian pesto call it pesto alla Genovese. The olive oil used for pesto, like the basil, requires a deft touch.
Ligurian oil is fruity and light, a far cry from the throat catching Tuscan oils.
Because pesto is not a cooked sauce, the quality of the salt is also important. Table salt can be too harsh. Coarse sea salt is preferred, but kosher salt will do in a pinch. Speaking of harsh, go easy on the garlic. Italian garlic tends to be smaller and less pungent than the garlic most Americans use.