If you live in Illinois and work in Missouri you need to file a tax return in both states. Your resident state of Illinois can tax ALL your income regardless of its source. Missouri can tax you as a non-resident on your earnings from work actually performed in Missouri.
On your IL tax return, you'll be able to claim a credit for the taxes paid to MO. Since Missouri's tax rate is a bit higher than Illinois', taking the credit should leave you with a small balance (if any) due to IL. (Note that, in TurboTax, you must complete the non-resident state return first, before you do your home state return, in order for the program to calculate and apply the credit correctly.)
If you wish, you may also increase what you pay to IL by making quarterly estimated tax payments to IL. This reference gives the details on that:
https://www2.illinois.gov/rev/individuals/filingrequirements/Pages/estimatedpayments.aspx
But, because of the credit, it probably won't be necessary for you to pay estimated IL taxes.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.