In that case, you'd better keep your passwords on plain paper in a heavy metal safe only – and never ever use a desktop OS to maintain/enter them. All desktop OSes known to me have "root access": it's called "root" with Linux/Unix, "Administrator" with Windows, and so on.
And like on all those: you should make sure what has access to "root powers" – and like on those you need to explicitly grant access. On Android, it's a core app often referred to as "SuperUser" managing that: as soon as a (new) app requests root access, you'll see a pop-up asking you whether you confirm (you can then chose yes/no and whether to always apply that decision when this app asks again).
Yes, like on all those, an application with "root powers" can (at least theoretically) access all your stored passwords, as any other data. That is why a properly coded application never stores your passwords in "plain text", but rather encrypts (or at least hashes) them.
Not rooting your device only takes that decision from you at the cost of not being able to "fully use your device". But yes, you should always be careful deciding what permissions you grant to an app – and in the first place, what apps you install.