Hiya!
Online banking is safe, secure and depending on which bank(s) you are with, feature rich.
It's taken quite a few years to get to the level it is now, which is long after the US and Canada took to online banking by the mainstream
There are some 'good practices' you can put in place to minimise any data security risks
1. Ensure your Internet browser has a high 'Cipher Strength' - I only know of this in Internet Explorer - Help>about internet explorer option. Make sure it's at least 128 bit. I think IE7 comes with 256 bit. This means for secure connections, you have the highest encryption standard - I think the US military have for years been using 256 bit.
2. Ensure your PC/MAC has suitable security software. Many MAC users don't run security software, before going online to your bank(s) website, research which ever browser you have or if it's Safari and it's security strengths etc. I'd still suggest to run 'some sort' of security software on a MAC, as it's only going to be a matter of time till the vrius and malware writers start to target MAC operating system in the future. For PC users, a good firewall, upto date anti virus, and malware detectors and removers is good to ensure you keep any nasty scripts off your machine.
3. Look for the Padlock or secure connection notification in your browser - it means you have a secure connection and the browser is 'correctly connected' - look for the same whenever you shop online as well - ALL reputable online shopping sites use secure connections. Also you may want to manually type in the bank's online banking secure URL and/or use their website to click on their links to get to the login page. Don't be tempted to click a link sent to you in an email to login to the bank - it could be a 'phishing' site.
4. Don't be tempted to keep your password and memorable info and login name/details on the PC itself - if you must keep them electronically, consider getting a USB stick and holding on that only - plug in when needed, get the details and then unplug it and store safely, or use a mobile phone memory etc - basically keep it on a system thats separate to the computer - best way of course is to keep everything memorised! but I know that becomes a problem with multiple banks and information. Look into the programs which securely store ALL your passwords on the computer and are accessed via one 'master password' - they also are well tested and secure offering 256 bit encryption - the likes of symantec offer solutuons like this.
5. Login regularly - this is to be able to check for any fraudulent or unrecognised transactions that may occur. The quicker they're spotted the better the banks chances of closing access and/or investigating.
6. Make use of the other features. By logging in online, you can setup allsorts of useful stuff, bill payments and direct debits etc. If you then use Quicken or Microsoft Money you can download statements and use the software to keep an accurate track of everything
Lastly - from experience after working at Intuit and being involved with the Quicken product - there are some banks out there who have excellent user friendly systems and others that aren't so great in other areas. I've lost count the number of times I'd call *&(&)^(%*%* bank to talk to their developers and get a 'we're not offering that yet' on ther online banking.
Overall, you're very very safe when using Internet banking so long as you are security aware.
Hope this helps!
Cheers! DtM! West London & Slough UK!