Blackwell, 25, was taken to hospital after Saturday's British middleweight title fight with Chris Eubank Jr was stopped in round 10.
He suffered a small bleed on the brain but has not required surgery.
"I'm told the vital signs are good and doctors are cautiously optimistic," Costello told BBC Radio 5 live.
Blackwell is heavily sedated, but doctors are gradually reducing the amount of drugs over the next two to three days.
"There are no major issues within the context of his condition," added Costello.
Towards the end of the fight, Eubank's father - former world champion Chris Eubank Sr - told his son to punch Blackwell's body rather than his face.
The ringside doctor then halted the fight in round 10 because Blackwell's left eye was swollen shut.
He was taken from the ring on a stretcher while being given oxygen.
Eubank Sr later told BBC Sport he would have stopped the fight had his son been in the same situation, while Eubank Jr said he eased off in the final round and that referee Victor Loughlin should have ended the bout sooner.
However, Eubank Jr's trainer Ronnie Davies disagreed and said both Loughlin and Blackwell's corner were right not to pull him out of a "title fight".
The British Boxing Board of Control said it is satisfied with the way the fight was handled and respected trainer Adam Booth told BBC Radio 5 live that Loughlin had "acted immaculately".
Meanwhile, boxing promoter and former world champion Barry McGuigan said it was an "isolated incident" and "nobody was to blame".
Nick Blackwell's family expect the boxer to wake from his induced coma in the next 48 to 72 hours, reports BBC boxing correspondent Mike Costello.