Do ensure that the online bank you're doing business with is covered by the competent authority who insures all savings accounts,current/checking accounts, and CD deposits up to a limit..
Do ensure that when you go to the online bank's website that you're on the real website. Believe it or not, Internet criminals have designed websites that appear to be similar to the banks website in hopes of tricking you into getting your personal information. You can confirm the genuineness by finding "a Lock" icon before your site address and also "https" as the first letters of your site instead of "http".
Do look over all your bank statements, credit card statements, etc., to be sure everything looks right and is legit. Talk to your bank immediately if you find any errors or see something that is questionable.
Do keep your password and account log in information safe and secure. Never share this information with anyone, especially through an email. Also, try to never use any common passwords like birth dates, names, or social security numbers.
Do be sure you use an anti-virus protection software on your computer at all times. It's also a good idea to back-up your files on a regular basis. You really don't want to take chances in this area.
Do log out completely from your bank's website when you've finished with your transactions.
Do change those auto-generated passwords you are given when you first open up an online bank account.
Do find out all the information you can about your online banks security. What kind of protection do they offer? How does that line up with other banks?
Do Immediately change your passwords if you have accidentally revealed your credentials.
Do use Two-factor PINs wherever they are enabled by the sites. This gives more security for your banking.