The answer is not very safe at all. Most 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, and it's a very weak link indeed.Most personal computers are shot through with security holes. This is especially true of PCs running Windows. The big problem is that if a hacker breaches the security on your computer, they can access your Internet bank account through it and pretend that they're you. 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.
If you decide to take on the security challenge of Internet banking, here are some important tips:
If you have a computer at work, running on a big network, it's highly likely that it's much safer than your machine at home. Its level of security can be measured (roughly) 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 (not just because they're against the rules but because your computer physically won't allow them) then it's likely that your security at work is very good. You may decide to use this computer for all your Internet banking (assuming, of course, that you've asked your employer for permission).
The only downside to this is that your network manager can spy on you and even collect your password as you type it in, but the chances are that they're earning plenty of money themselves and don't need to steal yours.
However you access your account, make sure you use a good password. 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).
If you bank through your home PC, you'll need to learn more about security. It's pretty much 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 really need Windows 2000, and even that still has a few security holes. Windows NT is pretty good once a few service patches have been added.
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. If you don't have that knowledge, you're probably better off waiting until the banks get their acts together. The way forward is for them to supply their own software that you install on your own machine and use for accessing your account. Only then will Internet banking be relatively safe for people without computer expertise.