Java is fairly secure, but only install programs you need
As briefly as possible: Java is a software development and deployment platform. You need it only if you run applications that require it.
The security flaws in Java are, in my opinion, greatly over-hyped. Other software people use on a daily basis, such as Firefox, Chrome, Flash, Windows itself, and really pretty much all software routinely have similar security vulnerabilities and receive regular updates. Most people don't bother to read what the updates are, but if you investigate the release notes or associated articles the next time you have updates pop up for Windows, Firefox, Flash, or pretty much any other program, you will see that these sorts of vulnerabilities, although certainly undesirable, are not at all uncommon.
In fact, in my opinion, it speaks well of Java's security that the media do make such a big deal when vulnerabilities become known. Microsoft releases updates for Windows every single month. Security updates are still released for Windows XP every month even though it has been in the field for 12 years now. The media doesn't bother to report such things because they are so routine, but no less serious.
As a rule, however, only install software you need. That doesn't just apply to Java; it's a general rule. If you don't need it, don't install it. In security this is referred to as reducing the attack "surface area." Less software means less possibilities for exploit, regardless of what software that is.
If you need Java, any programs that require it will let you know. You don't have to worry about things mysteriously failing. Most applications will clearly state somewhere whether they require Java. At worst you would get a "missing plugin" placeholder in most browsers.
Also note that Java works in two different scenarios: regular desktop applications and as a web browser plug in. If you have desktop applications that require it but you do not need it in your browser, you can keep it installed but disable it in the browser for improved security. This can be done in the latest Java versions in the control panel. Just go into Windows Control Panel, click Java, then on the Security tab un-check "Enable Java content in the browser."
Most of all, check for and apply updates promptly. That goes for all software. I strongly recommend setting the update option to either automatically install updates or at least automatically check and notify you when there are updates. That is one of the most important things you can do with regard to security, and that goes for all software.