A survey conducted by the German Marshall Fund asked respondents from Euro area nations what their desirability to intervene in Syria was. This same survey was also done in 2012, and it’s quite clear that opposition has grown since then. Not a single nation’s respondents were more opposed to Syrian intervention than they are now.
62% of those in the U.S. are now against Syrian military action, but that is actually near the bottom of this group. In Slovakia, Romania, and Portugal, over 80% of respondents expressed a desirability to avoid the conflict entirely.
An important note though is that polling for this survey was conducted between June 3 and June 27, so opinions may have changed.