Kobayashi overcooked his approach into the pit box as he came in for his second stop, hitting three of the crew ready to change his front-right tyre and the front-jack man.
Two of the team were taken to Silverstone's medical centre for treatment, one for an injured thumb and the other for cuts and bruises, whilst a third suffered a bruised foot.
A repentant Kobayashi, who finished just outside the points in 11th, said: "I want to apologise to the team.
"Of course, I had to push hard for points, but without doubt I braked too late in the pit lane.
"The front wheels locked, I couldn't control the car, and this is how I overshot the pit box.
"I am terribly sorry this happened and I hope the three mechanics I hit get well soon."
The trio are now back with the team and should be fit for the next round at Hockenheim in Germany in a fortnight's time.
Team principal Peter Sauber said: "I am obviously hoping our three mechanics make a speedy recovery from their injuries and are fully fit again very soon."
Given the unsafe nature of the pit stop, the stewards later fined Kobayashi 25,000 euros (£19,830).