Avoid holding cash above deposit insurance limits. You can play the account titling game as well as use multiple institutions. There are some banks that promise to spread your funds to multiple banks to get coverage; but you need to be sure you don't use those underlying banks elsewhere, etc.
For stocks and things, you've got some options. For best protection, you would want to have your stock ownership registered on the books of the company; you might be able to make that happen with a full-frills brokerage, ask about holding stocks not 'in street name' or you might have to transact via the registered transfer agent. Expect that to result in paying comissions and hassle.
If you don't want to go that far, making sure you don't have a margin account is a good step. Brokerages generally have to maintain customer deposits and holdings separate from the brokerage's propriatary holdings, but not necessarily for customer's margin deposits. Anyway, one should always be careful with margin.
If a brokerage fails due to their own poor investments, it shouldn't impact your holdings. Of course, if they fail due to poor record keeping or fraud, your holdings may not actually exist. SIPC provides some insurance that may apply, but the limits aren't very high and there's not that many credible brokerages to spread your holdings among.