Does it matter if the green card sponsor has criminal record?

For a family based permanent resident petition it does not matter that you have committed a crime in the past or even served time in a jail.  If you did your part by either paying a fine or going to jail or being under probation or whatever punishment the judge gave you, right now you are free to pursue your own life.  You can get married to an overseas citizen or sponsor other family members as allowed by law.  If you are eligible in all other respects like any other citizen, do the proper paperwork, there is nothing to fear.  Just remember that the USCIS is always trying to detect if an immigration petition is fraudulent and it might look more carefully in your application just to make sure that you are not planning to break the law yet once again, but nothing bars you fro moving on with your life.