The most important thing that you can do is to pray for them. Our heavenly Father loves them even more than you do and He desires their affections and works constantly to draw them unto Himself. Second Peter 3:9 gives this encouragement: “The Lord is not slow in keeping his promise, as some understand slowness. He is patient with you, not wanting anyone to perish, but everyone to come to repentance.” Knowing that the Lord is pursuing them, you need to pray that they will be open to the Holy Spirit.
Aside from praying, there are some other things that you can do. It is very important that you live as a witness for Christ. People need to be able to see Christ in you by your lifestyle, your moral standards and your compassion. Also, you need to be ready to be used by God when the time is right. The Lord will lead you when it is time to share the Gospel or describe the coming end times. Just be prepared and follow His agenda, not your own. “Let your light so shine before men, that they may see your good works and glorify your Father in heaven” (Matt. 5:16).
Your question was referring to the fear that they would miss the rapture. That would not be the worst thing. Worse than that would be if an unsaved loved one got hit by a bus tomorrow, sealing his or her eternal destiny. At least if he or she misses the rapture and is left behind, he or she will still have an opportunity to confirm all that you have been telling them –and have the opportunity to accept the Lord.
The New Testament is our covenant with God under the blood of Christ. In it, we are instructed about living a moral lifestyle and serving others so that they may know God. Galatians 5:14 says, “The entire law is summed up in a single command: ‘Love your neighbor as yourself.'”