There is something of an assumption in the question that immigration policies are either very open or very closed and thus perplexing for a distinction to be made by countries between open for immigration through legal, procedurally fair channels and closed for illegal trafficking.
Blunt fact is most of the planet is poor, governed by corrupt oligarchies and by most metrics has a very rough 21st century ahead of it. Since it is manifestly impossible for any country to absorb all the people that would like to live in a better part of the world if they could, each country - hopefully as a democratic consensus by its existing residents - establishes a procedural process with quotas of how many people it will accept for various reasons.
Prime Minister Turnbull mentioned in his phone call to President Trump that being visibly strict on illegal trafficking ("boat people" in Australian Media parlance) is to encourage applying through appropriate channels and to maintain faith that everyone you meet on the street is a legitimate resident with all the rights they are entitled to.
The ugliness of the off-shore processing "solution" is partly deliberate (i.e. to make the illegal route less desirable than the legal one) and partly ham-fisted bureaucratic indifference. The treaties that 1st world countries signed in the 20th century to pressure democratic development in emerging economies have become something of two-edged sword in the 21st - since how do you discourage illegal immigration whilst preserving the dignity of those involved? I don't envy politicians finding a middle ground in this quagmire.
Why Syrian refugees specifically? The general consensus of the West is that the vast majority of Syrians were consumed by a civil war/regional proxy war they didn't create (only the young and the foolish start violent revolutions). So taking in Syrian refugees is a more sympathetic choice than economic migrants or political refugees from never-ending hell-holes.
The United States is an example of how not to do immigration policy (is that controversial?). No side would agree the current arrangement is healthy or allows for procedural fairness. No one should ever fear being dragged out of their home and separated from their kids. The hard decisions have to made before it gets to that point. An economic migrant won't decide against their own interest; and rarely thinks about whether queue jumping undermines those in greater need.
We may eventually have a border-less planet; and that will be a good thing. For the foreseeable future though, economic migrants will always be ranked lower than refugees and refugees will ranked according to how novel the level of suffering in their origin country is - not an absolute measure of suffering (otherwise everyone from North Korea should get an auto-visa).