The early church lived in a time when social security and hotels did not exist. Since they do exist in our time, does that mean we can ignore verse 13? Obviously, the answer is no. Paul mentions two areas of concern: one is participating in the needs of Christians, whether local or elsewhere, and the other is providing hospitality to Christians on the move, perhaps on business or serving an ecclesiastical role such as Paul himself engaged in as an apostle.
It is interesting that Paul expected these two areas to be provided by all Christians. One reason for doing so would be that once again they would be expressing the reality of the divine family relationship, that they understood the importance of brotherly love. We should keep in mind that the needs are not only physical or financial. We can imagine how something like loneliness or desire for company would need to be dealt with by Christians, whether the recipients were residents or visitors.
Moreover, engaging in such expressions of love is an effective public witness. While we do not do them in order to be seen by others, it is inevitable that others will see us engaging in such practices. And they will see when we don’t. After I became a Christian, I discovered that Christians in different places were keen to help me. I also recall the intrigue that my workmates had when they discovered that I had friends in countless places throughout the country in whose homes I could stay and in which I was gladly welcomed. They never met my friends, but they were aware of what my friends were doing for me, and such expressions of genuine love made those non-Christians think about the cause of such love.
Of course, we cannot forget what Jesus says about brotherly care in the parable of the sheep and the goats in Matthew 25. The matters that are brought up in that description of the Day of Judgement are connected to various expressions of brotherly care and interest. He says in the parable that showing care to his followers is the same as showing care to himself, and he will show his appreciation by giving them not mere hospitality but a kingdom. Such will be the reward for genuine love expressed day by day.