Admittedly, Jesus never told his followers to pray for the sick (at least, not in so many words). But his brother did:

Is anyone among you suffering? Let him pray. Is anyone cheerful? Let him sing praise. Is anyone among you sick? Let him call for the elders of the church, and let them pray over him, anointing him with oil in the name of the Lord. And the prayer of faith will save the one who is sick, and the Lord will raise him up. (James 5:13-15)

Not only that, but Jesus gave his disciples quite a bit of instruction on prayer, much of which clearly applies to healing:

Pray then like this: Our Father in heaven, hallowed be your name, your kingdom come ... (Matt 6:9-10)

If you ask me anything in my name, I will do it. (John 14:14)

Whatever you ask of the Father in my name, he will give it to you. Until now you have asked nothing in my name. Ask, and you will receive, that your joy may be full. (John 16:23-24)

So why, someone might ask, didn’t Jesus and the apostles do that? Why do the Gospels and Acts record them speaking to sicknesses (“get up and walk!”) rather than praying about them? The short answer is: they don’t. They do both:

“Father, I thank you that you have heard me. I knew that you always hear me, but I said this on account of the people standing around, that they may believe that you sent me.” When he had said these things, he cried out with a loud voice, “Lazarus, come out!” (John 11:41-43)

But Peter put them all outside, and knelt down and prayed; and turning to the body he said, “Tabitha, arise.” (Acts 9:40)

And then there is the added observation that, while the epistles talk repeatedly about praying for everything - including, as we have seen, physical healing - they never tell believers to heal each other. (In fact, there are only five other references to healing in twenty letters; 1 Peter 2:24 is about the work of Christ, Hebrews 12:13 is about persevering in faith, and the remaining three are in 1 Corinthians 12:9, 28 and 30, in which Paul teaches that healing is a gift that not all have. That’s worth considering.)

So if someone tells you that you shouldn’t pray for the sick, but should speak to the sickness instead, you should probably ignore them. And then pray for them.