Your broker holds your cash and investments in trust. You're not investing in the broker themselves. So if they go bust, and they haven't been using client funds (which is against the law), you suffer no loss. If client funds have been diverted, then SIPC insurance kicks in up to their limits (which I believe are $500k with a $100k max for cash).
So in general, if you're not using some fly by night broker, you shouldn't be losing any sleep over this.