Generally speaking online banking is safe. Simply logging in to your banks' web site to transfer funds or pay bills should never have any security issues. But like all aspects of life there are certain steps you can take to protect yourself from nefarious individuals looking to scam you out of your money. It really boils down to remembering and religiously following the basics for protecting yourself online.
Strong Password - the majority of the account theft I see is typically due to poor passwords. You really, really want a strong password on your banking accounts. Yes, keyloggers can get 'em, but I'll say it again: the majority of account hacks that I see are simply due to poor password choices.
Don't Share Passwords - the number two reason I see for account theft? Telling your password to someone you "trust" who turns out either to be not so trustworthy, or simply not as cautious as you need them to be.
Email attachments - don't open 'em. At least not until or unless you're absolutely positive that they're expected and safe. Not positive? Don't open it. Just don't.
E-cards - as far as I'm concerned these are as bad as attachments. Don't send them, and don't open them unless you're positive it's expected and safe. (Hint: if it says it from "a friend", it's bogus). Not sure? Don't open it.
Phishing - never click on links in email messages from your bank or other official places. Instead, visit your bank as you would without email: type the link into the browser address bar and go there. Whatever it is they're supposedly emailing you about should be there. If not, they're either not a very savvy bank, or you just avoided being phished.
Secure your Wifi or shared connection - make sure that your Wifi connection at home has WPA2 encryption on it, and make sure that you take all steps necessary to use a public Wifi connection safely. If you don't it's often trivial for a hacker to sniff your log in credentials. In fact, whenever you share an internet connection, you may need to take the same precautions depending on your level of trust of your fellow users.
Never ever use a shared or public computer for sensitive work. You have no idea what's on them, what key loggers may be present, or what happens after you use them. Just don't.
One tool I suggest adding to your toolbox is the NoScript extension in FireFox. It disables Javascript (and flash, and more) and protects against cross-site scripting attacks for all sites until you say "this site is OK" - which is easy to do. It's a tad annoying at first, since you have to build up the list of trusted sites, but it's absolutely worth the additional protection.
My bottom line is that all banking, online or off, is dangerous. In my opinion, you're actually more likely to be affected by bad behavior you have no control over in the bank's back room than you are to have your keystrokes logged as long as you follow the simple security stuff you're already doing.
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