Online Casinos are very safe. The games are fair with very few exceptions. The key is choosing casinos with good reputations and/or reputable software.
At first, I was hesitant about online casinos myself. What would stop them from cheating the players. I was sold when I found out that the gambling software is controlled by the software companies. In most cases, the software is built by large companies that also program all the back end systems for the casinos. The casinos don't have access to changing the gambling programs.
Online gambling has been growing steadily since 1996 and many of the larger companies have sold their software to hundreds of casinos. Many of these companies are traded on major stock exchanges. They have invested countless millions on building and improving online casino software. Their software usually goes far beyond the software you see on the floor. The companies are experts at the latest internet security and office systems.
To the reputable software company, their reputation is of the utmost importance. These companies would not risk everything for a little extra profit. Remember that the casinos have the advantage in the first place and on some games like slots, the advantage can be substantial. Some smaller casino operations have been caught in the past. Most of them are out of business now, but a few of them are still around. These bad apples are well known by the operators of real online casino advice sites.
The good advice sites will then stop promoting these types of sites. Their business then dwindles fast. Casinos that don't pay their players are in the same boat. These players usually contact the advices sites to complain about non payment.
In addition, many casinos allow large, well known accounting companies to review their payouts. I personally don't get too excited about this (remember Enron) but it certainly doesn't hurt my confidence level.
Remember, the key is choosing casinos with good reputations and/or reputable software. Online casinos are safe if you play at a casino with good software or a casino that has an excellent long-term reputation.