I forgot the answers of the security questions. Not that I seriously need to do it now, but there might be a problem if I need to answer those in future.
I still remember my password.
How can I reset the answers of the security questions?
There is no way to reset your Apple ID security questions online after the Mat Honan hack and its aftermath.
I was able to reset my security questions by calling Apple Support on the phone, provided I could remember at least one out of three.
However if you can't remember any of the three, you might have trouble.
The reset procedure for security questions is here: https://support.apple.com/kb/HT201485
Below are more details and screen shots from a while back, but the links above have current information as well as discussion on how to get support in person and newer developments like Two Factor Authentication and using another trusted device.
If you know your password to your Apple ID and have set up a second email address, you can reset your security questions by logging into the web site https://appleid.apple.com even if you can't answer the security questions that normally are required to make changes to the existing security questions.
Once you have logged in, click 'Password and Security'. Then click/tap the link in the lower right that says Forgot your answers? Send reset security info email to m * * * *@monsters.edu
You will get an email with a link to reset not only the security questions and answers, but correct your birth date which can be used to verity your identity when you are calling Apple or trying to reset a password.
I have an Apple ID with a recovery email, so perhaps if you have not entered that second email, you might not be able to send the reset email and might need to contact Apple directly for help with your Apple ID. I know of people that have gotten things reset over the phone even after the Mat Honan incident where his account was compromised over the phone with Apple's assistance.
You don't need to remember the answers to your security questions to edit them. As long as you know the password to the account you can log at AppleID.apple.com at the Manage Account link and click on the security and passwords option on the left and change your questions. (I've just tested it on my account with no problems).