There is no guarantee for anything in life and financial stability of an insurer is one of those things, after recent banking crisis. But it would be taking things out of context if we compare them. Banks might have been taking high and unpredictable risks leading to the crisis.
On the other hand, usually the largest exposure a carrier has is facing unprecedented claims and they often re-insure against such risks. Also, they are more conservative in the way they invest money. They usually, buy properties and invest in type of assets that have regular income like rent.
One of the main reasons why they could go down is catastrophic claims. This is unlikely to happen since policies exclude biblical losses. In other words, if something that could have happened in thousand years has happened they could not have predicted it. Therefore, they can refuse to pay the claims on the basis that it was beyond the scope of insurance.
Although, it may not be reassuring since you may suffer large losses in such cases, it is not hard to understand that this clause is there to protect them from total wipe out. Generally, the responsibility falls onto governments if such a scenario was to play out. And people would probably still get some proportionate compensation.
Thirdly, every state in America oversees policy sales. So, car insurance companies are regulated by states and they have to apply for a license. In the same way, brokers and agents need to be licensed as well. There are certain barriers to enter into the market to protect citizens from rouge operators collecting premiums and disappearing. They apply certain capital requirements and tests. You should make sure that people you are purchasing coverage are licensed.
Also, each state offers insurance safety nets. In other words, they collect money (just like premium) from each participant and use it to pay claims if any of the licensed companies goes bankrupt. Generally, they are subject to some sort of regulation and some government body checks their financial soundness in other countries.
Naturally size may comfort people and therefore they may feel safer with the bigger names rather than the smaller and less known names. As mentioned above, there is no too big to fail anymore although governments are more likely to rescue them. With the same token, it would be easier to rescue a smaller entity and protect consumers. If you prefer to work with small and local ones there is no evidence that they are riskier.
Everyone has different preferences and take comfort from various things. It could be the fact that you recognize the brand or they are local. These are all valid reasons for choosing one over the others. In terms of security, there is never a 100% guarantee but it is as close to it as humanly possible.
There are companies that check financial soundness and publish reports but they usually only check the very large ones and therefore have a limited scope. Generally, people are not worried about purchasing coverage from them. Somehow we are programmed to worry about the sources we know little.