You tell them if they need to talk, or if there is anything you can that will help, they should let you know. You may also want to offer to help get their mind off things by going out and "seeing and talking about something different".
Other than that, you keep in mind that they may be someone who needs to process hurt alone and probably appreciates that you've expressed a willingness to help if you can, and trust that they'll ask for your support/help if/when they want/need it.
When people are the type who look to others for help/support in bad times, they'll usually ask (especially if someone has offered). As for those who prefer to process things on their own, they can find too much "pressure" to accept support and help intrusive.
So, I'd say, offer the help/support once, but offer ways for the person to get his mind off things as often as you want. He'll either welcome your offer, or appreciate but refuse it. In general, such an offer doesn't (I don't think) seem intrusive to most people who are going through a rough time.
Whatever you do, don't make the mistake that many people often do; which is to believe it "isn't healthy" for a person to want to process his own pain his own way.