Pew finds that there is very little public support in the U.S. even for “limited” strikes in Syria:
At least two things are worth noting about these results. The first is that Democrats are not rallying behind a president from their party on this issue. This is not what happened when Clinton ordered military interventions in the ’90s, but it is consistent with the reaction of Democrats to Obama’s wars in Afghanistan and Libya. Most Republicans might be expected to support hawkish measures, but on Syria they have been almost as reluctant to intervene in Syria as everyone else. While a slightly larger percentage of Republicans supports launching military strikes in this case, two-thirds of Republicans are opposed or unsure. Interestingly, 59% say that the U.S. should first obtain a U.N. resolution authorizing strikes on Syria, and even among Republicans there is a majority in favor of getting a U.N. resolution (54%). This is certainly not going to happen, but it reflects a somewhat surprising level of support for adhering to international law.
The public understandably fears the dangers of an anti-American backlash and the potential for pulling the U.S. deeper into the conflict. Finally, there is not much confidence that the strikes will have the desired deterrent effect:
If Republicans and Democrats in Congress want to respect the wishes of their constituents, they will vote no on any Syria AUMF resolution that comes before them.
Update: An ABC News poll finds similar opposition to military action regardless of party.