If the applet you want to use requires Flash then no, you have not made a mistake because it is what you need. As Braegnok pointed out, Flash has had a number of security vulnerabilities, which is one of the reasons it is being replaced by other options, such as HTML5 when that is possible. It is also not a particularly speedy program either, which is another reason it is being replaced.
As far as being in danger, yes, having Flash installed and running makes you vulnerable, but there are many ways to protect yourself. One is to make sure you're only visiting sites you can trust, so you are less likely to be exposed to any malicious code that can attack Flash vulnerabilities. Something else you can do, at least in Firefox (I don't use Opera) is have the browser set to ask you about running the Flash plug-in. If it is not running, then it will not run malicious code. (You might also want to look into this extension too: Click to Play per-element. If there are multiple things that want to run Flash on a page, this will make it so only those you click on will be activated. Firefox activates Flash, and other plug-ins, on entire pages at a time, and not as single elements) You can also look into using the NoScript extension, which will block various scripts from running on a page, again preventing/limiting when Flash is run.
Using an extension like NoScript may be overwhelming though, especially as some sites have a lot of scripts on them, and it can be hard to guess at first which you want to run and which you want to block. Just going with having Flash set to Ask to Activate though, and being cautious with the sites you visit will probably be enough and is not as burdensome. Of course that is my opinion and use a number of things to limit what scripts and content are run on webpages, so I may have a high tolerance.