No, I don't think most care about the refugee crisis beyond some lip service. The moment the time came to help the refugees, many displayed their true colors and considered the Syrian refugees fleeing a genocide (which many countries have the obligation to respond to due to the genocide convention that they ratified/agreed to) to be threats to them because of the actions of mainly non-Syrian non-refugees.
"But this is not just a problem of British tabloids. Those outlets, though loathsome, are in many ways just expressing an extreme version of the overwhelming sentiment among Western countries: that a single dead refugee child is a tragedy, but a million suffering refugees are a threat. Western voters have made clear to their leaders time and again that they want to keep refugees out, even if it means most of those refugees will die."
The drowned Syrian boy photo is viral social media at its most hollow and hypocritical
The pot has been stirring in Syria for decades because of that very Baathist "secular" and sectarian regime and not because of the US. One needs to just look at the uprising in Hama in 1982 when Hafez al-Assad (the Baathist) quashed it by killing hordes of civilians just like his son is doing today with the aid of a few countries (Iran, Russia, Lebanon, and Shi'a militias from elsewhere). The cause of the refugees today is, overwhelmingly, Bashar al-Assad (the Baathist) and his genocide of the Sunni Muslims.
