Before you publish a story on your blog or online, think about looking for a paying market for it and sending out some queries and submissions. Hunt around and bookmark places like RedInkWorks.com's Short Story Markets page, which has a ton of resource lists and market links, as do specific genre market sites like Ralan Conley's site for SF/F writers. Get into the habit of subbing a new market at least once or twice a month. Unless you're a Name, there isn't a lot of money to be made selling one short story, but sell five or ten of them a year and it starts to add up.
As for the stories for which you can't find a market, giving them away is better than letting them sit and gather dust. For one thing, readers really like free reads. I discovered this a few years back when I began posting an unpublished story on my old web site every month. As a gift to my readers each winter I'd make an e-book of the free stories from the entire year, throw in a couple I hadn't posted online, and give that away. At first I thought of it as an interesting way to promote my work, but it turned out to be much more than that for me.
For those of you who are not yet in print, some editors and agents are reading blogs these days to look for fresh voices. This is not to say that you'll get a book offer a day after you post a free story, but it's not unheard of for an interested editor to contact a blogger to see what else you've got.
You also can learn something in the process of writing short stories for your readers and getting feedback from them. A dozen of the stories I gave away on my web site eventually grew into or inspired new novels, which I turned around and sold to major publishers. The Darkyn novels, which are now my bestselling series to date, are based on three of those old freebie short stories.
If you're still looking for a way to post stories online and link or promote them from your weblog, and you don't care for anything I've suggested so far, you may consider using one of the story web sites like FreeStoriesCenter.com to act as your host, and then link to it from your blog. Before you use an outside host, make sure that you still retain all rights to your story, and the host is not profiting by selling subscriptions or downloads of your work. Free should really mean free.