A political solution would leave a portion of the Assad government in power, which is disagreeable for obvious reasons. Some analysts argue that strikes could help force the Assad government to negotiate a peace deal. The model here is the 1995 Dayton Accords, when NATO-led bombings in Bosnia helped force the government to enter peace talks.
What happens if we do nothing?
The likely outcomes of inaction don't look so different from the likely outcomes to limited strikes. After two-plus years of deeply entrenched civil war, not to mention active meddling from such regional powers as Iran and Saudi Arabia, Syria is being shaped by forces far more powerful than a few American cruise missiles. The fighting has taken on a self-perpetuating velocity that would be difficult to stop.
The biggest risks of inaction for the United States are that the taboo against using chemical weapons would become a bit weaker, especially within Syria, and that the United States would be perceived as further losing some of its international primacy. After setbacks in Egypt and tension with Saudi Arabia over the region's future, the United States would see its influence in the region take another hit.
The biggest upside of inaction for the United States would be to avoid getting sucked into a terrible conflict with no obvious solution or end in sight.
Would voting down the president's request hurt U.S. foreign policy?
Analysts tend to make this argument when they think Congress is about to block the president from doing something they see as worthwhile in foreign policy.
In 2010, when it looked like the Senate would refuse to ratify a nuclear arms reduction treaty that Obama had negotiated with Russia, a number of left-leaning analysts said that this would hurt U.S. credibility abroad by hobbling the president's ability to deliver on promises he makes to foreign leaders.
Plenty of right-leaning analysts warned the same thing when George W. Bush's foreign policy was challenged by a Democratic Congress after 2006.
The truth is probably that the American political process is not well understood in foreign capitals, just as Washington can often misread how Moscow or Beijing or Tehran make decisions. The United States, like any other country, is judged more on action than on rhetoric.