The question surmises it all. If my firefox browser puts something like "..Gecko/xxyy Firefox/nn.mm.." in the USER_AGENT attribute, then from security point-of-view, isn't it an open invitation to the webserver: come and exploit all vulnerabilities in my xx.yy version ?
Many online users are know not to upgrade their browsers for a long time (especially those using stable distributions like debian or centos). Here, they are openly exposed to any known vulnerability being exploited in their outdated browsers.
Some years back, you could have argued that the web-server might output a compatible content based on the browser type (IE/Firefox/Safari/etc..), but with the standardization of HTML,CSS,etc. I don't thing this argument holds now in 2013. If anything, this adds one more hole in a plethora of complexities that already need to be dealt with in network security. Don't you think so?