I'm sure this image is a real dinger at preschool, but otherwise there are only two kinds of people who think it (and the hundreds like it) says anything worth repeating:
The NativityJoseph and Mary were faithful Jews traveling from one part of a Jewish country to another part of the same Jewish country. They weren't leaving their homeland for foreign parts. They were actually returning to their ancestral homeland for a high holiday called Sukkot, which means "tents" or "temporary dwellings". (The Romans were smart people. They recognized that the best time to make all the people travel somewhere was when they were all going to be traveling somewhere anyway.)
When Joseph and Mary arrived at Bethlehem, their destination, they didn't demand special treatment. They didn't demand that the locals cater to their religious and cultural preferences. Actually, the people who lived around Bethlehem had the same preferences as Joseph and Mary, so no demands were necessary.
When Yeshua (aka Jesus) was born, Luke says that they laid him in a manger (probably a feeding trough) because there was no room at the inn (Greek kataluma, which is also translated "guestroom"). If this actually means that the local inn was booked up and the young couple had to find a place in the barn, it shouldn't be much of a surprise. Very likely every inn in the country was full because potentially hundreds of thousands of people were on the move. That's similar in a way to the current crisis, except that the mood would have been festive instead of threatening and fearful.
On the other hand, I'm not so sure that there was such a thing as an inn, at least not like what we think of. It's much more likely that they were at the home of a relative who was entertaining whole branches of the extended family, including great grandpa Shimon and great grandma Esther, and they were all vying for the only guestroom. I think this is kind of cool, actually. One of the rules of Sukkot is that you are to "dwell" in a temporary shelter (called a sukkah) for 7 days. Rabbinic tradition says it's sufficient to eat your meals in the shelter, but maybe this was God's way of setting Joseph and Mary straight. He arranged that the young Yeshua would keep the command perfectly, even when it was completely out of His control, by forcing his parents into what was probably a hastily constructed shack. (Probably not a stable like you see on postcards and certainly not a cave.) They laid him in a manger instead of in a bed because all of the beds were already filled far beyond capacity. They made do with whatever was close at hand and would keep the newborn off of the ground.
They weren't desperate, they weren't running, and they weren't foreigners. They were Jews visiting Jewish relatives in a Jewish land.
Does this mean that we shouldn't help desperate foreigners running from trouble? Of course not. It just means that the meme is based on a false premise.
The RefugeesI'm certain that many of the "Syrian refugees" are actually Syrian refugees. I'd want to get out of Dodgistan if I lived there too. Unfortunately, many of them are not Syrian and many of them are not refugees. They are Iraqis, Somalis, and a mix of other Middle Eastern peoples.
Of those who are actually coming from Syria and Iraq, many are military aged men. Those are not refugees. They are cowards who have fled the field of battle and possibly abandoned their families or they are infiltrators planning to wreak havoc behind enemy lines. We saw a very clear example of this in Paris recently.
Such terrorists--aka invaders--are probably a small minority within the mass of the refugees, but how much feces is acceptable in your ice cream? Maybe you prefer your vanilla with a little predigested Middle Eastern spice. Who am I to judge?
Even so, a few hundred or a few thousand terrorists really aren't the biggest problem because the real enemy isn't ISIS. The real enemy is Islam. All of it.
If you aren't a complete, blithering idiot, you have probably noticed that the West maintains a very high standard of living compared to other regions of the world, and that has absolutely nothing to do with the residual effects of western colonial imperialism or any other self-loathing nonsense your sociology professor might have fed you. It has everything to do with historic Western culture and Judeo-Christian religion.
Democracy (for what it's worth), emancipation of slaves, individual liberty, constitutional government, electronics, the Internet, automobiles, the printing press... these are all the fruits of Western Civilization. I don't care that some other civilization might have invented them given enough time. The fact is that they didn't. We did, and I like all of these things.
Islam, on the other hand, produces nothing but misery and death. Every Muslim country is a violent, squalid hell compared to the poorest Christian country in Europe. I would choose to live in Moldova or Romania 1000 times over Saudi Arabia or Iran. (Isn't it interesting that all of the very poorest countries in Europe have had significant Muslim influence? The further you get from Islam, the better things get.) Every tree can be known by its fruit, and Islam is among the most poisonous in the world. Everywhere Islam goes, Syria and ISIS follows by one name or another.
Allah hates all mankind and Islam is the proof.
We don't object to opening our countries to "Syrian refugees" because we're stingy or xenophobic. (We in the West are among the most generous and accepting people in the world.) No, we object to mass Muslim immigration because we're sane, because we don't want Syria in our own backyards. We don't want our economies wrecked, our industries crippled, our wives and daughters raped, and our children enslaved, and far too often, that's what comes with Islam.
I want to help the people who are suffering in Syria and Iraq--we all do--but importing them en masse into our own countries is suicide! That doesn't help anyone. It just turns our own lands into the same hell that they're trying to get away from. I admire your urge to ease the suffering of people you don't know. That is truly a godly attitude. But we must find a way to help that won't destroy the very thing that makes us able to help. Because, in the long run, the only thing that will significantly ease suffering in Muslim countries, is for Muslims to stop being Muslims.