There are allot of countries in the world that can offer asylum, assistance, or a place to live. America doesn't have to be everyone's "one stop shopping" for a decent place to live.
If anyone wants to take in, or individually sponsor a family they are free to do so. Just go to the US Government and tell them what you want to do as a private citizen. You can sponsor folks. But, that's a personal choice for you and your family. You are free to spend you income any way you wish, and open up your home to anyone you feel comfortable with.
At some point your value based decision is driven by or effected by economics. Your personal wealth and your square footage of space to house and take care of folks is a limitation, even for those of vast wealth, they too are limited. Heck, Sally Struthers didn't sponsor the entire Continent of Africa, and even Oprah has limits.
Interesting though.....Most folks offer support to where the support is needed. Build a school, support democracy, dig a water well, doctors without borders, various charities. We, as a country do our fair share. Time for the rest of the world to step up.
The fleeing Syrians could be supported somewhere in the middle east. The Venezuelans could be supported somewhere down south. The Mexicans could patrol their side of the border. Not the USA's obligation to solve everyone else's problem...all the time.