I'm thinking of rooting my android phone (HTC Desire 300) and was wondering if doing so would affect the internet security with stuff like online banking. Is it still secure to do my banking on a phone that has been rooted? Does it make any of my other online passwords or details vulnerable or unsecured?
When your phone is "rooted", applications can be installed (intentionally or otherwise) that can do almost anything on your phone, including breaching the Android application sandbox. Thus, by rooting your phone, you are removing a critical layer of the multi-layered security system. This can be mitigated, to a degree, by restricting root privileges to only applications that you trust.
Typically, to obtain root privileges, you have to run code for which you don't have the source code. This also lowers your overall security.
In the end, some people will choose to not root their phones because the risks outweigh the benefits, while others come to the opposite value-based decision.