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".

However, Inverness Caledonian Thistle and Greenock Morton were knocked out despite winning and finishing second in their groups.
Inverness beat Forfar 2-1 and Morton were 5-0 winners over Edinburgh City.
Elsewhere, St Mirren beat Airdrieonians 5-0, Falkirk defeated Brechin City 3-0, Motherwell edged Berwick Rangers 1-0 and Raith Rovers beat Cowdenbeath 2-0.
Falkirk, Motherwell and Thistle's group rivals Livingston had already secured second-round places as group winners while St Mirren, Airdrie, Brechin, Berwick, Raith and Cowden were unable to go through before playing their final group games.
In Group H, Premiership Partick Thistle were kept out by League One Stranraer until the 87th minute when Elliott struck from inside the box.
Thistle had beaten St Mirren 5-0 last Saturday and the Buddies recovered from that loss by putting five goals past Airdrieonians in Paisley.
Former Aberdeen striker Cammy Smith and ex-Hearts forward Gavin Reilly each scored twice for Jack Ross' side and Lewis Morgan got their fifth.
In Group A, top side Falkirk went ahead through Lee Miller's strike with Joe McKee and Alex Harris also on target against Brechin City.
Inverness CT went behind against Forfar Athletic when Marc Scott shot home but strikes in quick succession by George Oakley and John Baird turned the match in the Highlanders' favour.
In Group F, Motherwell had to wait until the 82nd minute to get in front of visitors Berwick Rangers, Elliott Frear netting after goalkeeper Robby McCrorie palmed out Chris Cadden's cross.
And at Cappielow, Bob McHugh and Robert Thomson got two each and Jai Quitongo also netted for Greenock Morton against Edinburgh City.
In Group C, Raith Rovers scored twice in the first half against Cowdenbeath, Lewis Vaughan and Liam Buchanan on target.
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.
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Partick Thistle beat Stranraer to progress in the League Cup as one of the best first-round group runners-up.