To answer you question directly, it is safe to install Java on you computer.
To answer your second question, is it safe to use http://multibit.org/ with Java for online transactions, the answer is Caution.
Do not get confused by the (JRE) the Java SDK and JVM ....and no need to try to learn them, since even Apple has just abandon it and Oracle (the maker) has troubles keeping it safe.
Now some experts will tell you it is safe if you keep your Java up to date. That is lots of garbage, since as you pointed out accidents did happen, so it is clear that does not work. The software updates you get are mostly fixes and some are unfortunately after the damage occurred!
Since we are talking about your money here, I would say it is your decision to use it or not.
The real question is about the Mulibit client as you pointed out. Multibit is not the safest out there so consider using others. Multibit connects directly to the network, and there could be your problem. Means you would have to keep the Multibit up to date, but that is no quarantine either.
...There is also more specific website for this type of question here:
Just to add my personal advice, I would caution you with such products, unless you are expert in keeping your computer safe.
It is your money!
So, after you have been told by some software geeks in here it is safe:
April 17, 2013.. Oracle issued a massive update to its highly targeted Java software, fixing 42 Java vulnerabilities, including 39 serious flaws that can be remotely exploited by attackers.
The Java fixes were part of more than 100 security updates issued by Oracle across its product line. The Java updates in Java 7 Update 21 can be used by cybercriminals in a variety of attack scenarios, including drive-by attacks if a victim visits an infected website.
Oracle said 19 of the flaws are extremely critical, carrying the maximum score in the Common Vulnerability Scoring System (CVSS), the standard scoring system used by many software vendors for assigning a severity level to coding errors. Security experts say it won't take long for malware writers to develop an exploit targeting some of the flaws.
Lets use StackEchange as an Example.
After you decided to trust them and turned java on (so you can use the site) they abused that trust on April 1, by having a Java based POPUP in what some think been a great joke, but that some of us did not want to have. We were not given a proper warning nor an option to turn it off. Granted it was a Joke, but you get my point what can happen, so use it with CAUTION.