It’s an interesting question. I assumed they didn’t mean “using fear” in the sense of holding a gun to someone’s head and telling them they have to believe or I’ll shoot you or torture you in some way; but that the idea of “using fear” is meaning something like “you should believe otherwise you’re going to hell”.
Because while at one level these can seem similar (i.e. believe or there will be a consequence), there is a significant difference between them. The first is a consequence that we would be imposing on a person, which is not something which we have the right or authority to carry out (“’It is mine to avenge, I will repay’, says the Lord” Rom 12:19, Deut 32:35). Therefore trying to invoke fear of that would not be right.
But the second statement is different, isn’t it? The consequence of the second is not something we’ve made up, but the judgement of the Lord of the Universe. We are not going out of our way to instil or promote fear; but fear would be a natural reaction to the truth of the consequence.
People often quote “fear of the Lord is the beginning of wisdom” (Psalm 111:10, Prov 9:10) and then amend the meaning of it by saying that by “fear”, we only mean “reverence” because maybe they’re uncomfortable with the idea of being afraid. But I respectfully disagree… the Lord is frightening… the more we comprehend of his nature (and our own) the more we should fear his wrath and his character. It is only through understanding the truth of the reality and the consequence of our sin that we will turn to Jesus in hope for salvation, through the work of the Holy Spirit. Thank the Lord for convicting us so.
My other point would be that Jesus himself did not shy away from the truth of judgement as a consequence and therefore neither should we. John 3:36 “Whoever believes in the Son has eternal life, but whoever rejects the Son will not see life, for God’s wrath remains on them.”
So I think if I had to answer the question I would say… ‘No, fear is not a motivator we should use for salvation. But truth is a motivator we should use; and the truth is sometimes frightening.