Those who know me, know I am not a sports fan, but I am an observer of people. In sports, it is easy to see when someone has neglected the fundamentals. Their play is off, their attitude is bad, and the only way to correct this is to humble themselves and revisit the fundamentals. That is the idea behind 1 Corinthians 10:12
"If you think you are standing strong, be careful not to fall."
It is normally the little things that we don't take care of that lead to the greatest failures in our lives. The city of Sardis (one of the cities mentioned in Revelation) sat on a high mountain with sheer rock cliffs. Their walls were high and strong. Because of this, they felt they were an impenetrable city. They became arrogant and comfortable, so much so that they neglected the walls of their city. Without their notice, cracks began to form in these walls and over time, these cracks became large enough for a man to fit through. Soon while the city sleeps in their arrogance and comfort, a rival army climbed the cliffs and surrounded the city inside. When the people awoke one morning, to their horror, they had become overcome. This is what happens to us if we become arrogant, comfortable, and don't pay attention to the fundamentals of our walk with Christ.
The word "think" here refers to a way someone has an opinion about themselves. It has nothing to do with whether that opinion is true or not just simply what general idea is the person considers about themselves. We have to be careful not to compare ourselves to others but to always compare ourselves to Jesus. Doing that will keep our opinion in line with Christ.
His phrase, "be careful" means to continually be aware, to be watchful, ever vigilant. He is saying that regardless of someone's self-imposed opinion that they are solid be ever aware of the condition of your heart, attitude, and actions. We can "fall" if we don't watch. The Greek here means to fail and come to ruin, spiritually, mentally, emotionally, and/or physically. It describes the downfall of someone who formerly had a haughty opinion of themselves.
We must never become so smug, haughty, or complacent that we allow tiny cracks in our faith, attitudes, and actions to eventually turn into larger cracks that will destroy our reputation and God's destiny for us. As the old children's song goes, "read your Bible, pray every day if you want to grow." I will add to that because our culture has fallen so much because of complacency, we have allowed each generation to fall further away because of our choices, we must be in church. Our children need it to have a firm foundation. Sports, dance, and all the other things we feel they "need" to do will come and go in their lives, but a godly attitude and faith are for eternity. A lack of faith ruins a life for eternity.