How do Christians discern whether they are on the narrow path that leads to life or the broad path that leads to destruction?

“Truly, truly, I say to you, he who does not enter the sheepfold by the door but climbs in by another way, that man is a thief and a robber. But he who enters by the door is the shepherd of the sheep. To him the gatekeeper opens. The sheep hear his voice, and he calls his own sheep by name and leads them out. When he has brought out all his own, he goes before them, and the sheep follow him, for they know his voice. A stranger they will not follow, but they will flee from him, for they do not know the voice of strangers.” This figure of speech Jesus used with them, but they did not understand what he was saying to them. So Jesus again said to them, “Truly, truly, I say to you, I am the door of the sheep. All who came before me are thieves and robbers, but the sheep did not listen to them. I am the door. If anyone enters by me, he will be saved and will go in and out and find pasture. - John 10:1-9 

Jesus is the "strait" gate. Jesus is the "narrow" way.

Thomas said to him, “Lord, we do not know where you are going. How can we know the way?” Jesus said to him, “I am the way, and the truth, and the life. No one comes to the Father except through me. - John 14:5-6

To pass through into life one must divest themselves of all pretense regarding sin, righteousness, and judgement. One must utterly reject the pride of life which claims to somehow either deserve or renders unessential both mercy and grace. There is but one single, solitary way for any and every person to enter into life and that is through Jesus. He is the door. He is the way.

The only thing that will fit through this door is the uncovered man, wretched in his sin and utterly naked and excuse-less before the eyes of the Righteous One. What else can a man bring to God but a begging hand outstretched and imploring? Oh sinner, call on the name of the Lord Jesus Christ and be saved!

Having thus come, desperate and helpless to be cured of sin, the gracious savior does not turn away or quench the bruised reed, the smoldering wick.

All that the Father gives me will come to me, and whoever comes to me I will never cast out. - John 6:37

In Christ transgression is covered. In Him the power of sin in the flesh is broken, not removed yet but broken, and no longer reigning as master. The Holy Spirit of God takes up residence in the newly cleansed tabernacle of the body and begins to conform the newly redeemed to the image of Christ. This activity of God will be carried out unto completion (Philippians 1:6). In Christ one is born again...born from above and they have eternal life.

For as the Father raises the dead and gives them life, so also the Son gives life to whom he will. For the Father judges no one, but has given all judgment to the Son, that all may honor the Son, just as they honor the Father. Whoever does not honor the Son does not honor the Father who sent him. Truly, truly, I say to you, whoever hears my word and believes him who sent me has eternal life. He does not come into judgment, but has passed from death to life. - John 5:21-24

That which was beforehand dead is now made alive. The question asks, "How do Christians discern if they are part of the privileged minority walking the narrow path to life?" This is tantamount to asking, "How does that which was previously dead know that it is now made alive?"

The purpose statement of John's gospel is that one may believe and, in believing, may have life in Jesus' name:

Now Jesus did many other signs in the presence of the disciples, which are not written in this book; but these are written so that you may believe that Jesus is the Christ, the Son of God, and that by believing you may have life in his name. - John 20:30-31

The purpose statement of the epistle of 1 John is that those who have believed may know that they have eternal life:

And this is the testimony, that God gave us eternal life, and this life is in his Son. Whoever has the Son has life; whoever does not have the Son of God does not have life. I write these things to you who believe in the name of the Son of God, that you may know that you have eternal life. - 1 John 5:11-13

If one has the Son of God (only begotten and equal as in John 5:18) one has eternal life and, according to 1 John, is capable of and expected to know it. It is apparently possible for one who has life to be yet unsure (or else why the stated purpose of 1 John). If one does not know that they have eternal life it is perhaps because they have not been well taught all that there is in Christ. 1 John is recommended reading for such.

As the Scripture says, however, there are many who do not have the Son of God and who, therefore do not have life. They have approached a misrepresentation of, a misunderstanding of the Son of God and have apprehended a savior who cannot save. They have yet to come to the gatekeeper, to the door of the sheep. It is entirely impossible for one who does not have the Son of God and, therefore, does not have life to know life as in 1 John. They have yet to enter the strait gate...they are on the broad path of the dead who think that they live and the only hope for such is to lift up their eyes, see the Son of God lifted up, and believe.

If I have told you earthly things and you do not believe, how can you believe if I tell you heavenly things? No one has ascended into heaven except he who descended from heaven, the Son of Man. And as Moses lifted up the serpent in the wilderness, so must the Son of Man be lifted up, that whoever believes in him may have eternal life.  For God so loved the world, that he gave his only Son, that whoever believes in him should not perish but have eternal life. For God did not send his Son into the world to condemn the world, but in order that the world might be saved through him. Whoever believes in him is not condemned, but whoever does not believe is condemned already, because he has not believed in the name of the only Son of God. - John 3:12-18

One may know that they are on the narrow path when one believes in the name of the only Son of God. This One who is begotten of God and who is one in being with the Father is the One who saves and is the One who sends the Holy Spirit to live within the believer. It is this indwelling Spirit that wrestles within us against the sinful desires of the flesh and bringing about conformity to Christ, thus testifying within us that we belong to God:

For all who are led by the Spirit of God are sons of God. For you did not receive the spirit of slavery to fall back into fear, but you have received the Spirit of adoption as sons, by whom we cry, “Abba! Father!” The Spirit himself bears witness with our spirit that we are children of God - Romans 8:14-16

Anyone who does not believe in, trust, and confess this Jesus as the only begotten Son of God cannot possibly have the internal testimony of the Spirit and so cannot possibly know that they have eternal life and will never be condemned. It is possible to be on the broad way and yet think the narrow is underfoot. The defining question is "Which Jesus do you confess?"