You can take a bus from the Denver airport to downtown Denver, and then from there catch a bus from downtown Denver to Colorado Springs. None of the buses are all that frequent, so depending on your schedule this may be easy or difficult.
Here is the schedule for the bus that goes from the Denver airport to downtown Denver:
http://www3.rtd-denver.com//schedules/getSchedule.action?routeType=9&routeId=AF
Now, to get from downtown Denver to Colorado Springs, you have two options. You can either take the "Front Range Express" bus or a Greyhound bus.
Here is the Front Range Express schedule. Note that it only runs on weekdays:
http://www.frontrangeexpress.com/?sid=1
Here is the Greyhound website. You'll have to type in your destinations and times to find a bus that might work. Note that there are a couple of different Greyhound stations in Denver, so make sure you pick one that is downtown, within walking distance of where the airport-to-downtown bus will drop you off:
https://www.greyhound.com/farefinder/step1.aspx