I live in St. Louis, the easiest and less expensive way to get there is just get on the Metrolink at the airport. It will take you right to the hotel at Union Station and should not cost you more than $2 each. It is safe and easy! Just hope you do not have too many bags. The Union Station Marriott is very nice and close to so much. Enjoy the Gateway City!
I use to live in St Louis and I would suggest the metro. Train will pick you up right at the airport and have you at union station in about 30 minutes. $2.00 one way. Also while you are downtown they have a free ride zone. This is 8 or 9 downtown stops that you do not need to pay to ride. Good luck.