I think people in the US, being ignorant of names such as these, would automatically pronounce it Bass-eel or Bas-ile. They will try to use phonetics. If you were to spell it
Basil though I think you would get the Baz-uhl pronunciation. I personally would pronounce
Basile with a French accent, as Basz-eel-uh (considering it is a French name, and the last uh would be almost indistinguishable). That is probably indistinguishable though as to actually how it is rolling off my tongue. :)