The Blades have won just one of their last five games and are now five points outside the play-off places.
"We can only say sorry to the fans, it was a bad performance. I think we are letting them down a bit too much at the moment," Baxter told BBC Radio Sheffield.
"I'm a football fan myself and losing a game ruins your week."
A first-half goal from Craig Jones gave the Shakers, who lost 6-0 at Coventry on Saturday, their first win in four matches.
Baxter said his team could have no excuses for their first league defeat at Gigg Lane since 1952.
"These are the games that we have got to go and win to get up the table where we should be. It's very hurtful to get beat," he added.
"I'm not really sure what was missing, because we started the game well. They say goals change games and it did.
"Once the goal went in we were chasing the game but there's no excuses. It was a poor performance from us."
Sheffield United forward Jose Baxter apologised to the club's supporters after their 1-0 defeat by Bury.