Most crooks don't waste time with fake $1, $2, $5, or even $10 bills. As the denomination rises from $20 to $50 to $100, however, the outlook changes. The risk is particularly high with the $100 bill because it hasn't been redesigned since 1996. Bills from $5 to $50 received counterfeit-resistant technology between 2003 and 2008; you should be familiar with their colors by now. It's unfortunate that production of a counterfeit-resistant $100 bill -- the denomination most attractive to crooks -- is years behind schedule, a debacle for the Treasury Dept. If you travel overseas, you may find that some banks and money changers refuse to handle USA $100 bills because of the risk.
Other countries combat counterfeiting with a simple measure: as the denomination increases, the banknote gets physically larger. This prevents bleaching a 1 Quatloo note and then reprinting it as a 100 Quatloo note. The USA doesn't do this, but the new $100 bill -- if it ever enters circulation -- will offer other protections.
What does one do? The safest course is never to accept the current, non-colorized $100 bill or the old, non-colorized $50 and $20 bills. Read this flyer from Treasury. If you get potentially problematic bills at a bank, inspect them while you're still at the teller window.
Counterfeit Rolex watches, counterfeit art, counterfeit Viagra, counterfeit DVDs, counterfeit "certified" coins... the world is awash in this stuff. It's an ancient crime. Anything that increases the value of an item by a hundred-fold or a thousand-fold is a temptation for crooks.
P.S. $500, $1000, $5000, and $10000 bills were withdrawn from circulation in 1969, but they remain valid. If you deposit one, it will be shredded. Most $500s and nearly all larger bills have a collectible value far higher than their face value. Take them to a reputable coin dealer for verification and appraisal.