The Pakistan pace duo were ruled out through injury, but have been under a cloud since being banned and then reinstated after failing doping tests last year.
Neither had taken compulsory in-house dope tests ordered by Pakistani authorities for the entire World Cup team.
Both faced life bans for a second positive test, and experts had warned illegal substances may still be lingering in their systems.
Border said it was better for Akhtar and Asif, who have been suffering from knee and hamstring, and elbow problems respectively, to miss the World Cup starting in Jamaica in little more than a week.
"Players who have been selected from other teams for what is cricket's biggest tournament are there performing to the best of their ability and on a level playing field," Border told The Australian newspaper.
"If it were to be true that it wasn't an injury, and it turned out be something else, then it would be better for all concerned that they didn't make the trip.
"If it was revealed at some later stage they had taken an illegal substance, then the sport doesn't need another black eye.
"I would feel genuinely sorry for both the two young blokes if that were not the case."
Border, who captained Australia to a World Cup win in 1987, said he was prepared to give Akhtar and Asif the benefit of the doubt.
"There has been a cloud hanging over both of them for a good while," he said.
"It is to be hoped that they are truly injured and that it's not a repeat of their previous (drug) problem.
"But if it is the case, it would be sad on the one hand because I believe they are both quality cricketers."
Newspaper columnist Peter Roebuck also expressed relief that Akhtar especially had been ruled out.
"Thank goodness Shoaib Akhtar has been ruled out of the World Cup," he wrote in The Sydney Morning Herald.
"He will not be missed ... Far from sustaining a great tradition of fast bowling in his country, he has undermined it. Worse, he has taken emerging players with him. Now he has been exposed as a drug cheat. The game is up."
The Daily Telegraph said sources close to the Pakistan team believe the fear of Akhtar and Asif testing positive again had influenced the decision to withdraw them.
"It is understood the duo would have been targeted by Cup drug testers as soon as they landed in the Caribbean and fear of a major Cup drugs drama weighed heavily on the minds of Pakistani officials," it said.