The best method we've come up with is to count the number of security "pings" we get in one day. (Firefox pings a Mozilla server for security changes about once per day) and multiply that active daily users number by three to get active monthly users. (See John's blog post for more on this.)
Over the last couple of weeks, we averaged about 90 million of these pings per day. Multiplied by three, that gives us an active user base of approximately 270 million.