Before entering ARCA competition, Basham built his racing foundation in grassroots oval racing, competing in various dash car and street stock events at tracks including Salem Speedway and Brownstown Speedway. This local experience shaped his driving background before he attempted the more competitive ARCA Re/Max Series.
In 2006, Basham entered the majority of races in the ARCA Re/Max Series, primarily driving the No. 94 car for Darrell Basham Racing, the team owned and operated by his older brother. His debut season proved difficult: after failing to qualify in his first seven attempts, he finally made his official series debut at the second Pocono Raceway event. Starting 39th, he finished 31st after engine issues ended his race midway through.
The pattern of mechanical misfortune continued across his early starts. At the Illinois State Fairgrounds dirt track, driving a Pontiac, he started 35th and finished 34th after running only three laps due to rear end issues. His third start of the season came at the DuQuoin State Fairgrounds dirt track, where he again started and finished 34th, this time retiring after just three laps with handling problems. He failed to qualify for the remaining events at Chicagoland Speedway, Salem Speedway, and the inaugural race at Iowa Speedway.
The 2007 season followed a similar trajectory. Basham's first two attempts with Darrell Basham Racing resulted in non-qualifications. At Kansas Speedway he drove the No. 0 Chevrolet for Wayne Peterson Racing in his only start outside the family team, but again failed to qualify. Back with Darrell Basham Racing at Winchester Speedway, he qualified in 34th and finished 33rd after running only eight laps due to handling issues.
He failed to qualify for subsequent events at Kentucky Speedway and Toledo Speedway before returning to qualify at Iowa Speedway, where he started and finished 34th after five laps with handling problems. His final points-paying start came at Berlin Raceway, where he started and finished 32nd after completing just two laps before rear end issues ended his day. Three more late-season attempts all resulted in non-qualifications, leaving the Berlin Raceway race as his final appearance as a driver.
Throughout his time in ARCA competition, Basham filled a dual role, serving simultaneously as crew chief for Darrell Basham Racing while pursuing his driving career. This arrangement was a reflection of the tight-knit family operation that characterized the team.
Racing ran deep in the Basham family across multiple generations. Beyond his brother Darrell, Dugan Basham is the uncle of fellow ARCA competitors Jason Basham and Mike Basham, both of whom also raced in ARCA competition. He had a daughter named Peggy, who died in 2010.