I have no problem whatsoever with on line banking.
All of my savings, banking accounts and credit cards are managed on line and over the last 10 to 15 years I only had a problem with one credit card that was not down to a breach of on line security. I also do most of my purchasing on line without any problems and run all of my household and car insurances on line as well.
I do find it strange that some people will not bank on line from the safety of their own home but are happy to access ATMs in public. Similarly those same people are often happy to pass all of their credit card details on line to various companies to purchase products.
Clearly nothing is fool proof but the, minimal risk, in my opinion, is worth it for convenience and to obtain improved interest rates and cheaper prices.