Yes, you can track the check number, license number and date of birth of your customers who pay by check by creating custom fields on the check payment. Once added, those fields will appear seamlessly on the check payment screen in Springboard:
You can indicate whether or not these fields are required. (if required, you won't be able to complete the ticket unless something is entered in the field.) You can also report on any custom check fields you create in Springboard's reporting application.
To learn more about how custom fields work in Springboard, check out this article.