Stefano Pioli was sacked by the relegation-threatened Serie A club on Tuesday after just one win in nine matches.
The club's main supporter group were supporters of Pioli and Bologna were jeered off after the draw against Lazio on Saturday at the Stadio Renato Dall'Ara.
"We must do everything possible to satisfy our fans and win over their respect and affection," Ballardini said.
"We didn't create a huge amount but Lazio only had one scoring opportunity. For long spells Bologna did well, though we had a dip for 15 minutes.
"Lazio are a great side, so I see positive aspects of this result."
Bologna defender Cesare Natali was unsurprised by the reaction of the fans but said the players had to accept criticism.
Ballardini's men sit just two points clear of the relegation zone but their situation will get worse if Sassuolo or Catania win on Sunday.
"The coach had a good approach. We executed his ideas. The consternation was predictable," Natali said.
"We did not win but we have proven ourselves against a strong team. We were compact and have left little room for players with great qualities.
"We have to accept the reaction of the public, it's part of the game. It's not an easy situation, but we have to be optimistic and start the (second half of the league season) in search of positive benefits and of results."
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