Drive to Durango and follow signs to the Durango and Silverton Narrow Gauge Railroad. It's near McDonald's and has a large parking area nearby. Buy a ticket for the train that stops at Needleton and ride the train 2.5 hours (~30 miles) to the Needleton stop. The train will drop you off next to a suspension bridge that crosses the Animas River. From here, it's a 6 mile hike to reach Chicago Basin. Note: you can also take the train from Silverton and be dropped off at Needleton.
No train access in winter months.