A Flight Instructor reported a NMAC with opposite direction traffic when they entered downwind at a non towered airport.

Narrative

While conducting landing practice with a commercial student at BDU another aircraft entered the traffic pattern on the opposite runway direction; resulting in a near-miss situation (approximately 500 feet vertical separation).We were operating on Runway 26; which was favored by the prevailing winds; and were adhering to the published noise abatement procedures; i.e.; remaining close to the airport and climbing promptly to pattern altitude. We made standard CTAF position reports as prescribed in the AIM (Aeronautical Information Manual).While turning right downwind for Runway 26; another aircraft entered the left downwind for Runway 08; approximately 500 feet above the standard pattern altitude. Visual separation was maintained at all times. The other aircraft subsequently performed a 360 turn to rejoin traffic for Runway 26.After communication with the other pilot; they indicated they were unable to see our aircraft on ADS-B. Upon inspection; we noted that our transponder had been inadvertently left in Standby (SBY) mode. Once the transponder was returned to ALT mode; the other aircraft confirmed our ADS-B target was visible.Contributing Factors- Failure to switch Transponder to Alt mode during pre-takeoff procedures.- Opposite-direction traffic selecting non-prevailing wind runway.- High workload associated with flight instruction and pattern traffic management.Corrective Actions- Reinforced cockpit checklist use and transponder configuration verification prior to takeoff.- Reviewed AIM and BDU noise abatement and pattern procedures with the student.