Kelly was suspended for six games and Austin received a five-game penalty. Each player appealed their punishments, and they are eligible to play while their appeals are considered.
Major League Baseball announced the discipline Thursday.
Kelly, Austin, Red Sox manager Alex Cora and Yankees third base coach Phil Nevin also were fined. Four players on the disabled list also were fined for entering the field during the skirmish: Yankees pitcher CC Sabathia, and Red Sox infielders Xander Bogaerts, Dustin Pedroia, and Marco Hernandez.
In the seventh inning of New York's 10-7 win on Wednesday night, Kelly actually threw at Austin twice. The first pitch missed, but the second one drilled him in the back. The designated hitter then charged the mound, prompting both benches to empty.
The Red Sox were upset with Austin after he slid late into second in the third inning and his spikes caught Red Sox second baseman Brock Holt, who had leaned away from the bag after receiving the throw from third baseman Rafael Devers for the forceout.
The Red Sox said Ramirez had a contusion on his wrist and X-rays did not indicate a fracture.
Ramirez was plunked with one out in the first inning. Boston fans booed when Ramirez was hit, but players from both teams remained in the dugouts and there were no confrontations. Boston's training staff spent several minutes attending to Ramirez before Cora replaced him with Mitch Moreland, who took over at first base in the second.