Nope, Syria has never ever been part of our sphere of influence, and same with the U.S who never had any influence in Syria(even today we and they don't have much, apart from giving some limited support to rebel groups fighting dictator Assad).it was a former French colony, and then ruled by a tyrant dynasty the Assad familly since it's independence from France to this day. it's only after the civil war, that we somehow reluctantly got involved, Syria has never been crucial for us anyway, never cared much about the country(it doesn't have much to offer anyway) and has been Russian/Soviet ally. Our focus has been on Gulf and other the middle eastern countries .
Who told you we don't have influence in the countries we have military bases? You think a country/power will come out of nowhere and establish a military base/presence in a foreign country without exerting any influence whatsoever there? Lool Good one. If that was the case then even Spain or Turkey will have a base in Russia.
http://www.huffingtonpost.co.uk/mark-curtis/britain-oman_b_11426144.html
Just a small example. just because we don't go around shouting/singing praises doesn't means you should think otherwise.