Do you think Civil Affairs Active and Reserve will be called to support the Syrian refugees when they eventually are allowed to come here?

If it were to occur that CA got engaged in this, it would be outside US soil due to posse comitatus rules.
One could make a fairly compelling argument that some engagement to support the existing camps in Jordan would be a good use of economy of effort assets, as some of the mission subset would involve staunching the outflow of certain groups because of persecution, improving conditions in the camps, identifying candidates for asylum in willing host nations, and identifying who the bad actors are. Those are all things that the UN is either not very good at or not doing altogether.
This is what we as Civil Affairs can contribute. Will we be committed to that task?
Not likely.
Force protection would be a real problem, especially if you go small, as I expect. Identifying bad guys implies some means to detain and question. That would mean the kind of agreement with the Kingdom of Jordan that the Obama Administration has eschewed. Improving camp conditions means a pretty significant monetary commitment, which neither Congress nor the administration seems to have any stomach for.
Then there is my observation that "solving" the problem doesn't seem to be the goal. Running out the clock is. This is not an issue that is going to just go away. Bosnia-Herzagovina is the most recent analogous operation, and that took the better part of a decade.

I wouldnt think there would be a need for military assistance for this. What's different from other countries is we arent talking about the situation where thousands of refugees show up at one time at our doorstep and the government has to scramble to find a way to house them and process them. The plan is to process them through the long standing refugee resettlement process. Plus, the numbers are not gigantic as some sources have stated.

Currently the US resettles around 70k refugees from around the world each year, and has for decades. Refugees are sponsored by approved resettlement organizations. Each refugee family is met at the airport by a representative and taken to an apartment already arranged for them to live. There are no tent cities or need for military barracks such as when thousands show up at once. Last year, of the around 70k refugees, less than 3k were from Syria. Sec State Kerry annonced that the US planned to up the overall worldwide refugee numbers from 70k to 85k in 2016 and 100k in 2017. Also, Pres Obama gave State guidance to accept at least 10k from Syria.

But as you can see from the numbers, it doesnt represent a huge increase in numbers, and they are allowed in as arrangements are made. So it is unlikely there would be a sudden influx that would overwhelm the current system. Some people may be concerned because certain republican candidates have talked about Obama admitting 100-200k Syrian refugees. But thats a false number.

LTC (Join to see)
What are the chances we would encounter problems similar or worse than those of the refugees of the Mariel boat lift? If we are looking at something similar to this then I would say, yes.