The answer is not very safe. Internet banking is automatically conducted over a relatively safe kind of Internet connection called Secure Socket Layers, and the banks themselves have high security which is rarely breached, but the weak link is your own personal computer.
Most personal computers are shot through with security holes. This is especially on PCs running Windows. The big problem is that if a hacker breaches the security on your computer, he can access your Internet bank account through it and pretend that you are banking. The bank won’t know the difference and you’ll find it very difficult to persuade them that somebody else transferred all the money out of your account, even though the transaction was conducted on your computer using your telephone and your Internet connection.
On the other hand if you have a computer at work, running on a big network, it’s much safer than your machine at home. Its level of security can be measured by how strict the limitations are on its use. For example, are you able to download software from the Internet and install it on your machine? Are you able to view all kinds of files on the Internet, including Java and streaming content? If you can’t do these things then it’s likely that your security at work is very good. You may decide to use this computer for all your Internet banking.
Whenever you access your account, make sure you use a good password preferably alphanumeric. Never, ever, store an important password in the Password List of your computer. Anybody with a decent knowledge of computing can read stored passwords in a couple of seconds (honestly, a competent eight year old could do it).
It’s essential that you run anti-virus software and a firewall to keep out hackers. You’ll also need the latest version of your operating system. Don’t do Internet banking on Windows 95. It’s too insecure. Windows 98 is marginal, you require Windows 2000, and even that still has a few security holes. Windows NT is the best among other operating systems.
Macs are relatively secure, though recently they’ve been attacked more often. Again, you need the latest version of your operating system for maximum security. Each time a new version comes out it plugs more of the leaks in the previous effort.
If you’re running Linux, it’s likely that you know a fair amount about computer vulnerabilities yourself. You’ll also know that Linux can be made safe. You’re off to a good start.
The bottom line on Internet banking right now is that it’s not safe. To get to a reasonable level of security you need a good knowledge of computers. The way forward is for them to supply their own software that you can install on your own machine and use for accessing your account. Only then will Internet banking be relatively safe for people without computer expertise.