To Examine, To Check
する verb
けんもんする
警察が私の車を検問している時、どきどきしました。
While the police were inspecting my car, my heart was racing.
You examine the problem. In order to do this, you need to inspect it or to examine it. And if you're inspecting people like this, you're probably the police.
This is a jukugo word, which usually means on'yomi readings from the kanji. If you know the readings of your kanji you'll know how to read this as well.