Blockchain.info keeps your wallet file which contain encrypted private key/seed, but they claim they don't have access to user's wallet and it's been proved with their open-source software even though i think there's no sure way to verify if blockchain.info use exactly same software within their website/app without add any backdoor.
As for other private info they store, i think only your email address, IP/browser details & mobile number (if you add one).
If you have privacy concern, i suggest you to use Desktop wallet, whether it's SPV or full nodes wallet such as Electrum and Bitcoin Core.
Even if they don't have the access on users wallet then we cant still be sure even they do claim it.This is why I don't really trust out online web wallets but somehow I do see blockchain is a good one as long you do know how to set security features of it to avoid on getting hacked but speaking generally with your keys then its not possible to posses those informations which isn't surprising for a web wallet.
Of course we can't trust them by what they say, thief don't tells you that he/she is going to steal you something. Of course blockchain.info makes daily or weekly backups to keep everything safe and in need they will have access on your wallets I believe but why will they use your wallet? I believe they won't because they don't need it and won't steal you anything.