No, wars have been shown to help the economy rather than hurt it. The airstrikes in Syria will only affect stock prices for companies who mass produce weapons and other necessary things for military purposes. The economy in the United States has been stable enough lately that it will not significantly impact stock prices.
At the end of the day, most people in the West simply don't feel connected enough to Syria for it to make much of a difference. Companies with significant stakes in the area will no doubt see their share price drop, but the majority of companies will carry on as normal because it simply doesn't affect them.
As airstrikes in the Middle East become more and more common, their impact on stock prices becomes less significant. While people will be watching these latest airstrikes carefully, business will go on as usual. A large terrorist strike on the U.S. would have a significant impact on stock prices, and by taking pro-active measures, the U.S. will hopefully succeed at disabling the terrorist networks.
No, the air strikes in Syria will not have a significant impact on sock prices, because there has been war in the world for 12 years now. We have not had peace since September 11, and the stocks have had to learn how to adjust. We will not notice a big difference now, either.