Unfortunately, each browser comes with it's own set of security problems. The companies that make the browsers routinely put out patches, but these patches need to be installed automatically in order to take effect. In some cases, such as Firefox, there are significant problems in pushing down patches.
However, some of the problems with browsers have nothing to do with the browsers themselves, they have to do with us, the users. It is very easy for our computers to become corrupted simply by visiting a web page. That is, we don't have to download anything or run anything or install anything, we just need to visit a web site and our computers are compromised.
Now, what does "compromised" mean? Well, that covers a wide range of evils. Some Internet badness (aka, malware) deletes all our files. Imagine losing all your pictures, all your music, all the programs you paid money for and now have to repurchase just because you went to what you thought was safe website.
Other malware turns your computer into a tool that is used to attack everyone else's computer, sometimes to send out SPAM. (Of all the SPAM you've ever received, it's likely that 99% came from someone's home or work computer that was infected with something.)
Worst of all may be the current trend to try to retrieve personal information.
Consider what would happen if the compromise gave the bad guys your banking information. This does happen, regularly, and the banks will usually do nothing about it; it was your ID, your password/PIN and perhaps even your computer that was used to transfer all your money to a numbered account in the Caribbean somewhere.
Or if they learned enough from looking at your email and your files. They could probably apply for a duplicate birth certificate and with that, a duplicate Driver's License. At that point, they are you. Maybe they only apply for credit cards in your name (which you are on the hook for), maybe they want a passport in your name.
All of the above is not only possible, it happens, every day, many many many times a day.
Currently, the safest browser, in terms of both security updates as well as the ability to protect us from ourselves is Google Chrome, and so Chrome is by far the recommended browser for most web surfing.
However, there are some business related sites that simply do not work properly with Chrome, and for those sites, Internet Explorer must be used.
Click here for some more information on Google Chrome, as implemented by the ITDS support group.