Name and address
Your name and your address are the only two things you should have pre-printed on your personal checks. This information is required by your bank and any business that accepts checks. In fact, many businesses will not accept starter checks that do not have this information pre-printed (as opposed to handwritten) on the check.
Many businesses require your phone number on your check before they'll accept it. Remember that every piece of information you put on a check is one more piece of information that can be used by an identity thief to craft an identity based on you. Luckily, revealing your phone number does not put you at much risk, so while you shouldn't have it printed on your checks, it's OK to write it on your check if requested.
Your license number
Your driver's license number is a number that is unique only to you. As such, you should keep it secure. If it falls into the wrong hands, it can be used by identity thieves who will use it to "prove" they are you. If a business asks for your driver's license number on your check, ask if you can provide a phone number or some other identifier instead.
Social Security number
Do not write your social security number on your checks. No business or agency, other than the U.S. Social Security Administration or the major credit reporting bureaus (TransUnion, Equifax, or Experian) should ever request your social security number.
Account and routing numbers
At the bottom of your check are the routing number, which identifies the institution you bank with, and your account number. This information is all an identity thief needs to take money from your account; however, that's the information a business needs as well. Because of this, it's best to not use checks at all---if you can avoid it.
Every time you write a check, you are giving away all the information needed to access your checking account and the funds in it. A much safer option is using your credit card (not a debit or check card), because they offer consumer protection. For paying bills, avoid sending a check through the mail each month by using direct debit through your bank.