How do we resubmit a green card application (through marriage) after receiving a rejection letter because one of the checks had an incorrect date?

You should resend everything with the correct the check and cover letter, and put the rejection letter on top so that the U.S.C.I.S. cashier can see that you are responding to a rejected application. The file should be coordinated with the missing items by U.S.C.I.S. when you send everything back. Due to the limitations of the Lawyers.com Forums, Alan Lee, Esq.'s (the "Firm") participation in responding to questions posted herein does not constitute legal advice, nor legal representation of the person or entity posting a question. No Attorney/Client relationship is or shall be construed to be created hereby. The information provided herein by the Firm is general, and requires that the poster obtain specific legal advice from an attorney. The poster shall not rely upon the information provided herein as legal advice nor as the basis for making any decisions of legal consequence.    

Hello,   It would be best to start a new application with the corrections and replaced missing items. Good Luck.   Disclaimer: This information is of a general nature and is not, nor is it intended to be, legal advice. For legal advice about your specific situation, please speak to your immigration attorney.