Browsers are only software that sends requests to your website. They can be used maliciously or not. Your website should be designed in such a way that it does not matter what requests are sent, your site is safe. Anything that a malfunctioning browser can do, an attacker could do more directly. So, in a large sense, no, the vulnerabilities in browsers have no special effect on your website security.
That being said, there is a danger in having to provide for the lowest common denominator when considering security measures. A great example that is currently ongoing is that older versions of Internet Explorer cannot use SSL ciphers and protocols which are not vulnerable. Therefore, webmasters and web hosts are having to make a decision between being secure and supporting access from some of their visitors. Within the next month, there will be a big move away from the old protocols, hopefully forcing upgrades, but in the meantime, there will be hosts out there which choose to be less secure to provide for the needs of vulnerable clients. So, in this sense, they pose a threat, but not directly.