You can visit Qualys BrowserCheck to check whether your browser has any known security issues. It supports IE, Firefox, Chrome, and Safari, though you may need to install their helper plugin.
If you use Firefox, visit the Firefox plugin check web page. It can detect old plugins with known vulnerabilities and provide instructions on how to update them.
I recently ran across Is my browser up to date?, a very simple site that checks just the version number of your browser. I cannot vouch for it, and I don't know how good it is, but it claims to support IE, Firefox, Chrome, Safari, and Opera. It does not check for vulnerable plugins, extensions, etc.
If you want to check for old browsers with known vulnerabilities, you use Windows, and you are willing to install software on your machine, I can highly recommend Secunia PSI. It has a large database of applications and information about their versions (what is the latest version, which versions are known to be vulnerable). It will scan your hard drive and find vulnerable/outdated versions of software installed on your system. It can detect browsers and browser plugins with known problems. However, it is not limited to browsers: it also checks for insecure versions of other software.
This may or may not be what you are looking for, but it is worth knowing about.