I was talking about this last night with my wife and she mentioned she saw some statistic showing Christians were, until now, de-prioritized and actually given refugee status at a disproportionately low rate compared to their population in the ME. I don't have a source, but thought this might be useful to bring up. It sounded like this was why he did that.
Yes we should, and Muslim refugees are also victims of religious persecution. Unfortunately, Trump's order says we only care about your persecution if you're in the religious minority in your country.
But this type of vetting is used all the time. Refugees that come to the US (and other countries, including my own, Canada), examine refugee claims using very similar methods.
If you are a minority (religious, ethnic, political) that is used as evidence of persecution and helps your claim for refugee status.
One should be very careful in ascribing rational motivations to things the Trump administration does - even his rhetoric (vague allusions to the attacks of September 11, 2001) isn't borne out by the executive order. He did not ban immigrants from a single one of the countries of origin of the 19 attackers.
He said this because demonizing Muslims got him elected. If he were only considered about the alegged de-prioritization of Christians, he would say "we will no longer de-prioritize Christians".
If Christians were de-prioritized, it was done by the UNHCR, not the US. And there is no direct evidence suggesting this happened, the fact that 10% of Syria is Christian does not mean 10% of the refugees would be.