You can set label's border via its underlying CALayer property:
#import <QuartzCore/QuartzCore.h>
myLabel.layer.borderColor = [UIColor greenColor].CGColor
myLabel.layer.borderWidth = 3.0
Swift 5:
myLabel.layer.borderColor = UIColor.darkGray.cgColor
myLabel.layer.borderWidth = 3.0
Here are some things you can do with UILabel and its borders.

Here is the code for those labels:
import UIKit
class ViewController: UIViewController {
    @IBOutlet weak var label1: UILabel!
    @IBOutlet weak var label2: UILabel!
    @IBOutlet weak var label3: UILabel!
    @IBOutlet weak var label4: UILabel!
    @IBOutlet weak var label5: UILabel!
    @IBOutlet weak var label6: UILabel!
    override func viewDidLoad() {
        super.viewDidLoad()
        // label 1
        label1.layer.borderWidth = 1.0
        // label 2
        label2.layer.borderWidth = 5.0
        label2.layer.borderColor = UIColor.blue.cgColor
        // label 3
        label3.layer.borderWidth = 2.0
        label3.layer.cornerRadius = 8
        // label 4
        label4.backgroundColor = UIColor.cyan
        // label 5
        label5.backgroundColor = UIColor.red
        label5.layer.cornerRadius = 8
        label5.layer.masksToBounds = true
        // label 6
        label6.layer.borderWidth = 2.0
        label6.layer.cornerRadius = 8
        label6.backgroundColor = UIColor.yellow
        label6.layer.masksToBounds = true
    }
}
Note that in Swift there is no need to import QuartzCore.
See also
CALayermasksToBoundsCALayerSwift version:
myLabel.layer.borderWidth = 0.5
myLabel.layer.borderColor = UIColor.greenColor().CGColor
For Swift 3:
myLabel.layer.borderWidth = 0.5
myLabel.layer.borderColor = UIColor.green.cgColor
Swift 3/4 with @IBDesignable
While almost all the above solutions work fine but I would suggest an @IBDesignable custom class for this.
@IBDesignable
class CustomLabel: UILabel {
    /*
    // Only override draw() if you perform custom drawing.
    // An empty implementation adversely affects performance during animation.
    override func draw(_ rect: CGRect) {
        // Drawing code
    }
    */
    @IBInspectable var borderColor: UIColor = UIColor.white {
        didSet {
            layer.borderColor = borderColor.cgColor
        }
    }
    @IBInspectable var borderWidth: CGFloat = 2.0 {
        didSet {
            layer.borderWidth = borderWidth
        }
    }
    @IBInspectable var cornerRadius: CGFloat = 0.0 {
        didSet {
            layer.cornerRadius = cornerRadius
        }
    }
}
You can use this repo: GSBorderLabel
It's quite simple:
GSBorderLabel *myLabel = [[GSBorderLabel alloc] initWithTextColor:aColor
                                                     andBorderColor:anotherColor
                                                     andBorderWidth:2];
UILabel properties borderColor,borderWidth,cornerRadius in Swift 4
@IBOutlet weak var anyLabel: UILabel!
   override func viewDidLoad() {
        super.viewDidLoad()
        anyLabel.layer.borderColor = UIColor.black.cgColor
        anyLabel.layer.borderWidth = 2
        anyLabel.layer.cornerRadius = 5
        anyLabel.layer.masksToBounds = true
}
Solution for Swift 4:
yourLabel.layer.borderColor = UIColor.green.cgColor
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