How do I know if a potential new employee is a “long-term” unemployed person?

You could

• Ask them, bit facetious but actually a good starting point
• Find government departments and employment programs looking to place these people.
• Talk to shelters and voluntary groups who support these people.
• Talk to church or other community centers who know them and can vouch for them.
• Many are disabled or handicapped so there are support services and groups who are training and attempting to place "rehabilitated" people.

I think the below suggestions might be found helpful.

• Ask him his about previous employer and then confirm from there when he left the job.
• Track his personal accounts at social media and networking website. Review the posts/tweets on the walls. I think they should also give you an idea about his current or past employment history
• Moreover, this suggestion might be a little wispy but it works some times. Google the real name of that potential employee, if he has shared any bit about his working, job or employer anywhere on the internet, he can be tracked.

Ask them about what projects they are done and just look at the project and find about technical difficulties in that project from him and see his answer and you will get to know easily whether he worked on that particular project or not.