I think in general, absent any outside implications, most Americans could agree this was the right thing to do.
However, most Americans realize that this is not usually a one night stand where you go home with a smile on your face. This action inevitably leads to other actions, which inevitably lead to more involvement, more body bags, more dollars out the door. So the feeling is that they can't possibly rate this a good thing, because in all likelihood, it'll lead to things they *don't* want. (This is more or less analogous to being happy your teenage daughter is well adjusted and picked a good boyfriend, and then some months down the road finding out your new title is "Grandpa"). A number of polls rate this as historically low support (~50%) for a US military strike.
This is one area where the Trump admin will catch a break. Had the Congress given Obama the proper military authorizations some years ago as he requested, for good or bad, the US would already be neck deep in this war and would certainly own it. Trump would have basically been in the same position as Obama was in 09. He certainly has the freedom now, to figure out how involved he wants to get.
At least by keeping us out of it, there has been one somewhat fortunate side effect for follow on administrations (even though they won't admit it). With Russia in there, the US will never own all of Syria. That's gives us more flexibility to have more "one night stands" without having to own the war ourselves. (There was no power in 2003 Iraq, for example, who we could reasonably pawn responsibility off to or even point to as being "in the way").
However, the whole thing is a mess. There's nothing really pretty about any of the options.