An employee takes smoking breaks at work and I would prefer if they didn't. One concern is that they are asthmatic and this cannot be helping their health situation. Is there any legitimate incentive I can offer to stop them smoking?
Short answer: No.
If it's not illegal and they are not smoking where they shouldn't, then it's not your concern. It is their decision, knowing their own health issues.
Leave it be.
The only way someone can stop smoking is if they choose to do it themselves, not by someone dangling an incentive in front of their face.
As a smoker myself, if someone kept trying to get me to stop smoking, I would be extremely annoyed and it would only detriment my relationship with that co worker.
No, you can't. Nor should you.
Smoking is a (bad) lifestyle choice, which is not something you have any right or business being involved in. If they want to smoke themselves to death, that's their prerogative.
If you start down this particular rabbit hole, other employees could feel that this is unfair advantage. Or the smoking employee could feel discriminated. Or you could come across as overly pushy. There really isn't an appropriate way for you to get involved in this.
If you're the boss, on addition to setting "not here" rules (which will make them take longer breaks to go farther from the building), you can offer to pay for stop-smoking programs as part of the company benefits package available to everyone. You may be eligible for an insurance rebate or reduced rate if you do so; ask your carrier and/or shop around.
Remember, this is an addiction, even if the drug is legal. They can't stop without a major effort. Pretending that it's easy is not helpful or effective. Especially if you are in a part of the world where the general opinion has not yet turned against smoking.