Use a good anti-virus, have a good software firewall. Use a secure browser (NOT IE, unless you have no other choice). Do banking only from one – secured computer/location. Use wired versus wireless for that station/computer (or extreme wireless settings, including blocking all unknown MAC addresses: http://phoebusg.com/index.php?option=com_content&view=section&layout=blog&id=3&Itemid=9
Free anti-virus: free.grisoft.com
Free firewall (including some anti-malware functions): http://personalfirewall.comodo.com/
Following the above, then yes I would agree with the first poster – that it can be safer than using a credit card.
Most banks allow even for automated bill payment, or monthly/time interval based payments. Personally I get all my bills electronically, and then pay them electronically.
Your bank additionally can keep track of your usual activity, and to some extent give you protection against unusual activity (aka hacker moving funds).
Other thoughts. Make sure you see HTTPS in the address bar post-log in. Be careful with “email sent from your bank” – they should never ask you for your details over an e-mail.
Happy banking :)