If you apologize, you will be showing the person you like that you recognize your mistake, and that it was out of character for you. However, you should only do this if it was something extremely noteworthy. Oftentimes, the embarrassing event will have been much smaller than you think it is, and apologizing will only remind her of it.
Minimize the situation by explaining how it's not that bad.
This will show a certain level of confidence by taking responsibility for what you did and also showing that you're not particularly fazed by it. Even if you are bothered, if you act like you aren't, you will effectively deal with the situation by bulldozing it with your confidence.
Make a joke out of the embarrassing event.
Doing so can be effective because it shows confidence, shows ownership for your own actions and demonstrates an ability to look at the lighter side of things.
Ignore the situation.
This is often the best strategy. Most embarrassing events are only memorable to you. The reality is that most people can be too consumed in their own lives to notice your every single mistake.