The approved mRNA vaccines went through the same approval process as any vaccine. And once approved, they are monitored for safety like any other vaccine. Between pre-approval testing and post-approval monitoring, we would have detected any issues. So the proof is in the pudding — lots of countries have approved them and none have found risks that are worse than the disease they protect against (currently only COVID but there are more mRNA vaccines in the works).
There's also no reason to fear the way they work. Other vaccines introduce antigens (molecules that your body doesn't like and produces antibodies to attack) in various ways — sometimes with a weekend virus, sometimes with a dead virus, sometimes just the antigens themselves. mRNA is just another way to introduce antigens so your body learns to fight them. For a little while your body follows the instructions in the mRNA to produce the antigens, and then your body learns to attack those antigens. It's not all that different from the way other vaccines work. mRNA breaks down pretty quickly in your body so it's not even in your system for very long, and there's no mechanism in the body for mRNA to produce lasting changes. So it's a lot like you got a cold: for a little while the cold makes your body produce molecules, then your body fights it all off, and then in the end there's no permanent change except your body learned to fight off that particular antigen.