As I already wrote some months ago in a previous post dedicated to Mali (here), the main and objective interest of those allies is to remain passive, providing a symbolic and mainly mediatic support, that is showing off as supporters without bearing the brunt, might it concern security, military expenditures, or the possible loss of influence in case of failure (which nobody can wish).
Following the principles of economic science, positive externalities will benefit to the political (and passive) allies, whereas negative externalities will exclusively lie by the U.S.A and France. Mali alike, only two countries will pay for a global good.
1. The threat against their citizens could increase, as the allies of Syrian government could be tempted to back terrorist movements, tasked to retaliate against both allies, while up to now, nobody but islamists (the allies of Assad's enemies) have ever threatened Western countries.
2. France and the U.S.A show themselves as providing security to the rest of the world, while their assessment is that they are not threatened by the Syrian regime. Most of their traditional allies being rather in favour of a political solution, they will provide the minimum political support, that is at the lowest possible cost, whatever this cost is.
As war consists firstly in the commitment of military assets, it costs quite a bunch of money. Claiming their mission with regards to the world's security, France and the U.S.A are currently facing critical budgetary deficits and will pay in this affair for their own security, of course, but to the benefit of the rest of the world, including islamic countries, which are encouraging them to intervene, while being largely undemocratic...
3. A further concern is their strategy: what is the expected end state?
Up to now the ambitious plans for Iraq, Afghanistan, Libya have not proved particularly successful. Mali seems to be the only one remaining on good tracks (at least for the time being).
A probable scenario for Syria could be as follows:
Currently the islamist rebel groups are growing, apparently supported by Persian Golf monarchies, which are not a paradigm of democracy. Their extremism, their organization in local militias indicate that their victory could turn up to a bloody massacre against minorities and lead to the collapse of the Syrian state structure, ruled by the Alawite elite and the Baath party. Anarchy could then prevail for years, in one of the most sensitive regions of the world, at the borders of Europe.
In such a case the victory against the current syrian regime could turn out as a bloody defeat, which would directly lead to a large isolation of both France and the U.S.A if they would try again such an offensive against another country.
Current French loneliness in Europe with regards to Syria may be interpretated as the first sign of this weakness.
Regarding the U.S.A, their military and economic power will remain unchanged by a failure could bolster the isolationist trend, thus creating a vacuum in international relations.
On the long run, only a stabilized and democratic Syria could increase the credit of both countries. However none of them seems to be ready to invest the amount needed for such an project, frigtened by the previous failures.
Therefore, and this will be my conclusion, France and the U.S.A cannot expect any positive externality, or for the non-specialists of Economics, positive outcome) from next week's strikes.