Online banking itself is safe, and the banks will do all they can to keep their customers and to encourage them to use it. Including paying out should you be a victim of fraud.
Precautions should be taken. No-one will ever ask for your password so don't give it to them. It will be VERY rare to get an email from a Bank or similar asking you to visit a site and verify your details. If you do then the best thing to do is to go to the banks homepage using the URL that you know and login from their. DO NOT click and links given in any emails.
To should down your accounts would be over reacting.