Network security software has been elevated to such a level that it's probably more dangerous nowadays to use your credit card at dinner, or giving your card number over the phone to place a mail order. The quick answer to security fear is that online investing is safe. It's impossible to isolate yourself from the Net if you have any investments, a credit card, or a bank account. More than likely not. All stock transactions transpire in an encrypted medium.
Encryption is strong stuff.
No incidents of online brokerage security breaches have been reported in the media. You should see an unbroken key or padlock at all times when accessing an online brokerage account or placing a securities order. The question of network reliability, as opposed to network security, is far more difficult to answer. Network brownout or blackout caused by a regional power failure or switching disruption. A server crash and subsequent site failure.