The history of this can be traced back towards the French mandate rule of the region. After the fall of the Ottoman empire and the split of the Middle East through the Sykes-Picot agreement, the French ended up in charge of Greater Lebanon, the region containing present day Lebanon and Syria.

A common French colonialist tactic was to use a previously disenfranchised minority group as their chosen elite. This would ensure that the fate of the group would become tied to the fate of the French, as the group would have more to lose from changing than status quo than keeping it (and even though the mandates were not true colonialism like Algeria, similar control tactics carried over). The Alawis were the sect chosen in Syria.

So, throughout the mandate period the Alawis were given greater and greater status. They came to dominate the state apparatus and, importantly, become overrepresented in the Army. This is important post-independence as the army, and particularly the officers influenced by the Pan-Arab Baathist ideology, became increasingly politicized.

Hafez al-Assad, an Alawi air force officer, was one of these Baathist members. Following a failed coup in 1962, the Baath officers launched a successful coup in 1963 and began a "Baathification" of the country. This was then followed by 2 more coups: the first in 1966 by the Military Committee against the National Command in which Assad and a man called Salah Jadid took power. Then, in 1970, Assad overthew Jadid and became the sole leader of Syria.

Assad then set about making his rule concrete. This eventually developed into a state that can be seen to fit Ibn Khaldun's (d. 1406) theory on the rise and fall of civilizations. Assad created a state based around the 'Assabiyya (social solidarity/unity typically based on a tribal or clan model) of the Alawis and with the da'wa (ideology) of Baathist Arab Socialism. This made the Alawis an integral part of the Syrian state. Even more than under the French, they came to occupy almost all positions of power and army roles. Also like under the French, this linked the Alawis to the success of the Syrian state. Even as Sunnis (particularly those in the countryside like where we can see the 2011 rebellion began, less so those in the cities who benefitted from economic reforms in the 1990s) became increasingly disenfranchised and disgruntled, there was little that the Alawis could gain from changing the system.

Therefore, Alawis and Syria are in a form of Catch-22. Although Syria could possibly be seen under the "weak state" as defined by Salame, as the government is dependent on the consent of the Alawi class, its main bastion of support, the Alawis cannot remove this support without facing a radical lessening of their status and possible reprisals from the other religious groups in Syria.

I hope this helped to answer your question with a basic history of why the Alawis came to dominate politics! If you have any questions please ask.