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.