The person at Adobe was not very educated on the subject. Compared to newer verions of Java, yes, it's less secure. But the only real danger of Java (not JavaScript) is having Java enabled in a web browser. By default in Mavericks and Yosemite, Java is off.
If you install Java 8 from Oracle, it will automatically enable Java in your browser. You then need to go into the System Preferences Java control applet to turn it off.
The Java 6 runtime linked to above does not enable Java in your browser. It's there simply to support apps (like Adobe's) that need it to function. In that respect, it's perfectly safe. I have Java 6 and 8 both installed on my Mac. I need 6 for X-Rite's ColorPort app, and the newer one for Java apps I have to use to upload client images to get prints made. With Java disabled in the web browsers, it is, again, perfectly safe.