I do not think that the west generality does. Politicians are acting in the misguided belief that they are showing charity to fellow humans in need and need to show their society is not uncaring. I think they have difficulty weighing up the advantages of helping others against the peril they put their own citizens and society.
I suspect also they have no trust that neighbouring countries to trouble spots will be able, or wish to, uphold their responsibility to keep genuine refugees near their border until it is safe for them to return; and so the west will have a situation to deal with anyway. Of course the correct way to deal with that situation is to ensure strict border control and send folk to help the genuine refugees (until they show by their actions that they are not wanting help) at those neighbouring countries.