I won't get into whether Francis actually said these words or not. Rather, I will show in this blog post that the content of the quote runs contrary to scripture and, if carried out, would actually undermine the spreading of the gospel.
What is meant by the words "Preach the gospel at all times. If necessary, use words."? When people say this, they mean that you should let your life speak volumes. You should conduct yourself in such a way that people can tell that you are a devout follower of God. Your life will be a powerful witness to other people. Now, this part isn't contrary to scripture. Certainly, there's plenty of Bible passages that correspond to this view of letting your life be a witness. For example, Jesus said “You are the light of the world. A city set on a hill cannot be hidden. Nor do people light a lamp and put it under a basket, but on a stand, and it gives light to all in the house. In the same way, let your light shine before others, so that they may see your good works and give glory to your Father who is in heaven." (Matthew 5:14-15). And in 1 Peter 2:12, we read "Keep your conduct among the Gentiles honorable, so that when they speak against you as evildoers, they may see your good deeds and glorify God on the day of visitation." (ESV) In John 13:35, Jesus said: "By this all people will know that you are my disciples, if you have love for one another.” (ESV)
So, certainly letting our lives be a witness is a biblical teaching. The unbiblical part of this quote is the "if necessary, use words" part. Words are always necessary! Look, if you just conduct yourself in a godly way, but you never talk about Jesus and His love, do you know what's going to happen? People are just going to think you're a really nice guy. That's all. They're not going to say "Wow! Look at the kind of person Christ has made him! I would like for Christ to make me that kind of person as well!" No, they're going to think "Boy, Sam sure is a good fellow. I'm so lucky to have a friend like him." They will only think the former if you let it be known from where your moral strength comes from, and they will only know from where your moral strength comes from if you talk about it.
The Apostle Paul would never have taken this let-our-lives-be-our-sole-witness tactic. In fact, he wrote "How then will they call on him in whom they have not believed? And how are they to believe in him of whom they have never heard? And how are they to hear without someone preaching? And how are they to preach unless they are sent? As it is written, 'How beautiful are the feet of those who preach the good news!' But they have not all obeyed the gospel. For Isaiah says, 'Lord, who has believed what he has heard from us?' So faith comes from hearing, and hearing through the word of Christ." (Romans 10:14-17, ESV)
Paul thinks that words are necessary. And since his words are divinely inspired (2 Timothy 3:16), then The Holy Spirit thinks they're necessary too. And since The Holy Spirit thinks they're necessary, then so should we.