The traditional personal homepage counter is unprofessional, but a lot of large sites and commercial services use counters to show how popular they are (yes, it's the bandwagon fallacy, but it works).
Usually, it's something like
*Over 2500 design companies have signed up for our wireframing service.*
[logos of the biggest companies]
*278,046 users are currently signed on*
*56,218 galleries hosted.*
*200,000 videos posted.*
The first example shows that a service/product is endorsed by many in the industry, which could pique the visitor's interest and inspire confidence. The second would be useful on a social network, since social networks are useless unless they're popular. The third and forth show that a service is both popular and robust, as well as showing off how much content the site has.
Plain traffic statistics generally aren't worth advertising, but if your application has just grown from a small open beta to having 5 million hits a day, then you may want to celebrate that on your homepage for a few weeks.
One of the few exceptions is if your website is for a cause. Then a hit counter may be relevant and interesting to people. It's a good measure of how much support and interest the cause is getting.