What kind of bad things have you heard about Coinbase?

In general, the legal entities with good reputation like Coinbase and Bittrex, etc... can be trusted to follow their standard procedures. Whether these are sufficiently secure or not, is up to you to decide. My advice is to keep only cryptos that you are actively trading on these platforms and everything you are HODLing should be on cold storage. If you don't have access to the top trusted exchanges and have to deal with more obscure ones, then proceed with caution. Assume they could vanish at any points and keep your transactions small and spread across multiple exchanges, rather than all the eggs in one nest.