I agree with Archer that much depends upon the individual. So what is it that you are worrying about?
Are you worried about your equipment failing? If so and you are using well cared for equipment, don't. Equipment that is well maintained is unlikely to fail during your dive. If you don't own your own equipment yet, there is an opportunity to improve your comfort level. In order to relax during the dive, you have to put some trust in your equipment and the best way to do that is to use your own so that you know that it is well taken care of. Plus the more you use the same equipment the more familiar you will be with how to use it, which also increases comfort. Much the same as driving a rental car vs driving your own.
Are you worried about your skills? Practice and experience will help build comfort. If you aren't confident in the basic skills that you learned in your certification course, spend some time practicing them on every dive until you are. The more confident you are in your ability to handle problems underwater, the more you will be able to relax. Buoyancy control is the hardest skill to master and takes time and effort to develop, so if you are struggling with that you should actively work on improving and have some patience.
Are you worried about being attacked by some aquatic animal? Unless you are diving with great white sharks regularly this is a waste of energy. Very rarely is an injury from an aquatic animal the result of an unprovoked attack on a diver, most often it is the diver's fault by harassing the animal or by invading the animal's territory so that it feels threatened and attacks in response. This is true of all aquatic animals including predators that divers often fear like most sharks, barracuda, moray eels, etc. Maintain a respectful distance from marine life as well work on your buoyancy control so that you don't touch anything unintentionally and your chances of being injured underwater are extremely slim.
Are you worried about running out of air underwater? Your SPG tells you everything you need to know about your air supply, and as long as you monitor it frequently you should always know when you need to return to the surface. As long as you are diving in open water and using your dive tables or dive computer to properly plan your dives to avoid required decompression stops you should always be able to get back to the surface in 4 minutes or less if you need to. There is plenty of air up there. If your air consumption isn't very good right now, don't worry about it, it will improve with time. Don't let other divers (or your own embarrassment) pressure you into extending your dive into unsafe limits so that you are risking running out of air. Always plan your dive so that you are on the surface with no less than 500 psi (35 bar) of air in your tank. If you are diving with good buddies they will understand and be supportive, if not you should find new buddies. Bottom time can also be extended safely with a larger capacity tank (say 100 cu ft instead of 80) or by staying shallower.
Are you worried about diving maladies like air embolism or decompression sickness? As long as you follow your training by breathing continuously and by planning/executing your dive to remain within no decompression limits using your dive tables or dive computer you are at very very little risk of injury. You are much more likely to get injured driving to the dive site than you are during the actual dive itself. Besides following your tables or computer and making slow ascents, you can control secondary factors by staying well hydrated, avoiding heavy exertion, wearing proper exposure protection for the conditions so you stay warm, and avoiding diving when ill or under the influence of drugs or alcohol. You should also be aware that most cases of decompression sickness are not severe and if responded to promptly with 100% oxygen first aid followed by recompression treatment in a hyperbaric chamber it can usually be resolved without any lasting effects. The costs of such treatment can be expensive though, so it is always wise to carry diving insurance such as that offered by Divers Alert Network and others. It is inexpensive and well worth the peace of mind.
I hope this helps. If you are worrying about other things or I haven't convinced you that you shouldn't be overly worried about the things I mentioned, send me an email and we can talk more about it.
Good luck.