Today online security is a very important factor. Many businesses are completely based online, and there is tons of sensitive data available to check out only by using your web
If it doesn't affect many users, then I think notifying the site administrators is the most you can be expected to do. If the exploit has widespread ramifications (like a Windows security exploit) then you should notify someone in a position to fix the problem, then give them time to fix it before you publish the exploit (if publishing it is your intention).
A lot of people cry about exploit publication, but sometimes that's the only way to get a response. Keep in mind that if you found an exploit, there's a high likelihood that someone with less altruistic intentions has found it and has started exploiting it already.
Edit: Consult a lawyer before you publish anything that could damage a company's reputation.