Rooted phones are just as secure as an unrooted phones if you never grant root permission to any apps. The problem is that if you root your phone, you're bound to give root permission (otherwise, why are you rooting your phone in the first place), and applications that you give root permission may turned out to be rogue or leak their permission to allow an untrusted applications to gain root-like permission.
Running rooted phone is safe as long as you know which app to give root access and which are not. Problem is, even assuming that you only pick trustworthy apps they still can leak permissions inadvertantly (in security parlance, this is called confused deputy problem), so you must really be careful when choosing trusted apps.