First, you probably cannot avoid that situation entirely, but should rather be reassured that Chris' concerns were dealt with promptly and sincerely by the admins of the other site. Bottom line: it's probably more important how you respond if/when it happens, than spending too much time and energy worried before it happens.
Copyright in most countries is automatic and does not require formal process or notification — but to make it clear, it's worthwhile posting a copyright notice (ala. "Copyright Sagelady 2010") on all your pages as a reminder.
Most stackexchange sites also have a licensing page which describes how a third party may (if at all) reuse your content. Many of us use a Creative Commons license that allows some reuse, but often with certain requirements (ie. must be attributed, cannot be used for commercial gain, etc.). Have a look through the Creative Commons' site to get a feel for what kind of license, if any, suits your needs.
Attribution needs special mention, as that is the main area where the reuse of BasicallyMoney's content fell down: it was reproducued but without any indication (ie. attribution) of it's source. Many SE sites have specific attribution pages that describe this, so have a look at a few and you'll see what others have used.
User agreements — you should give some thought to how users are licensing their contributed content to you. From a legal perspective they own the copyright to their postings (questions & answers), although there is an inferred, but limited, legal right for you to use that content in some way (ie. displaying on your SE site). IANAL, but this all seems potentially very messy and theoretically a user could revoke that 'right' at anytime, requiring you to remove all their postings. Many sites make use of a user agreement, such as 'by using this site you are agreeing to...', which grants some kind of non-exclusive license to the SE site owner to use the content that users post.
So, there you go. You should consider:
- Copyright notices (add to your footer).
- Licensing.
- Attribution details to help people appropriately reference your content.
- User agreements.
At the end of the day, these are not going to prevent content theft but at least you are drawing a line in the sand from day one, and no one can later argue a misunderstanding. It will also make it easier to deal with the consequences of content misuse in an efficient fashion — as it seems occurred for BasicallyMoney.
The big picture here is that you should focus your energy on creating your own great content and inclusive community, rather than worrying about what the other guy is doing. No one can steal your community.