If you agree to accept a partial rent payment on the due date, put in writing what the arrangement is. State the amount of rent the tenant pays you (the partial amount), and the date that the balance is due. If you charge a late fee per your lease, also specify whether you will be charging the late fee and the amount, or whether you will be waiving the late fee. Sign this notice and have your tenant sign it as well.
In Lieu Of The Above, Always Fall Back on Your Lease
Your lease states when rent is due each month. If you do not want to accept partial rent payments, and feel as if you want backup, then include language in your lease that prohibits tenants from making partial rent payments.
Why it is bad to accept a partial rent payment - the reason it is not a good idea to accept a partial rent payment on the due date is that if your tenant can not afford the rent this month, they most likely will not be able to afford it next month. The longer you let this go on, the further behind they will become on paying you your rent.
Another reason not to allow a partial rent payment is, if you have multiple properties it is not a good practice to have different rent policies. If you let one tenant make partial payments and not your others, and your other tenants somehow find out, you could be accused of discrimination.
This Is Your Business
When tenants have financial difficulties, they probably need to prioritize which payments they must make and which they can slide on a bit. So, they might test the waters to see whether you will be the creditor they can put on the back burner. If you let tenants slide on the rent, then you will probably see partial payments and late payments regularly.
Remember, you are running a business and doing so requires that you have rules, and that your tenants abide by them or they will unfortunately need to leave.
Check Your State Law
In some jurisdictions (check your state law), if you accept partial rent payment you might not be able to evict your tenant despite what they told you. If you accepted an amount less than the full amount, you will waive your right to collect full payment for that month in some jurisdictions.