Major League Baseball said Wright tested positive for growth-hormone-releasing peptide 2 (GHRP-2).
Wright told reporters in Fort Myers, Florida, on Wednesday that he had failed an offseason test and appealed.
"We couldn't figure out how this substance got into my body," he said. "But at the end of the day, it's my responsibility. It falls on me."
Wright said he could not prove it was unintentional, so he accepted the suspension.
"It is what it is, and I've just got to move on from it," Wright said. "That's what's frustrating about it. I worked my butt off this offseason since getting the surgery. I feel like I'm in a good spot. From an offseason drug test, this comes up. I've been dealing with it.''
The suspension will begin at the start of the 2019 season. It also means Wright is ineligible to play in any postseason games this year.
In a statement, the team said: "The Boston Red Sox fully support Major League Baseball's Joint Drug Prevention and Treatment Program and its efforts to eliminate performance-enhancing substances from the game. While we are disappointed by the news of this violation, we will look to provide the appropriate support to Steven at this time."
Wright, an All-Star in 2016, started the 2018 season late after undergoing left knee surgery and serving a 15-game suspension under Major League Baseball's domestic violence policy. He made four starts and six relief appearances, went back on the disabled list in late June because of the knee, then returned in September.
In January, he and the Red Sox agreed to a one-year contract for $1,375,000, a raise of $275,000, in a deal that avoided arbitration.
Wright, 34, went 3-1 with a 2.68 ERA in four starts and 16 relief appearances in 2018. He was removed from the American League Division Series roster following the opener against the New York Yankees, a day after reinjuring his left knee during a workout. He had arthroscopic knee surgery Nov. 13.
"Obviously disappointed,'' Boston manager Alex Cora said. "It's something that we didn't expect. We'll adjust.''
President of baseball operations Dave Dombrowski said last month the team is unlikely to spend big money on a closer, which means the role is likely to fall to Ryan Brasier, Matt Barnes or Tyler Thornburg. Kimbrel remains unsigned.