That information is the only way to get money wired directly into your account so you don't have a lot of options. You should be reasonably comfortable giving out that information as there checks and balances (as noted above) but more than this the banks tend to err on the side of avoiding a PR nightmare if someone uses routing/account to defraud their customer.
For bank security you should be more concerned about
a) your credit card, only use secure https sites and ones you can see are dealing with lots of customers
b) your identity, someone with your social security number, a recent bank statement and some basic information about you (like family, birth location etc) can assume your identity
c) your bank login, be sure to create a strong password, pref 10 characters or more with numbers, symbols and upper & lower case. A site like http://strongpasswordgenerator.com/ can help here.