First of all, we must note that it is not at all clear that President Obama does support the idea of American military strikes against Syria. He has resisted ordering such strikes for quite some time now and has been reluctant to declare that Syria used chemical weapons since that would put more pressure on him to order strikes. That said, there are clearly people who do think that the US should use military force against Syria.
One reason for this would be that some people think that the US has an obligation to fight for what it perceives to be justice and for international law. If we allow Syria to get away with violating international law and using weapons of mass destruction, we will be reducing the important of international law. This would, in some people’s views, reduce the security of the US because international law would have no meaning and countries would be more aggressive about doing things like getting weapons of mass destruction.
Another reason for ordering military action would be that it would help to have more control over the situation in Syria. If we do not participate in that conflict, whoever wins will not owe us anything at all. We might have very limited influence with them. That might lead to a situation where they would do things, like siding with terrorists, that could be harmful to the US.
No one thinks that Syria poses any direct and immediate threat to us. However, there are those who think that military force will make us safer because it will help us to avoid long-term, indirect threats.