Answer:

“So dear Christian people if you've got a human heart, let's look at the thing right. Now, taking medicine or anything hinders your faith, keep away from it. See? But I will tell you... Now, I know you said, "Well, I know doctors that laugh at Divine healing." I do too, and I know preachers that laugh at it too. That's right. Many of them laugh at it. Frankly, to be honest with you and my Bible, I found more unbelieving preachers in Divine healing than I have doctors. 'Cause many, many, fine doctors comes to my place to be prayed for. And we set right down in the room, heart to heart, and talk. I never have heard a one yet, but what didn't tell me, "Why, Brother Branham, I believe." And especially a surgeon, said, "I've laid people out," and said, "why they just quit breathing. We'd take them down and after while they start; there they go again, breathe and get well. What did it?" He said. I said, "God." He said, "You're right." Said, "We might move an appendix that's ruptured, but we can't heal the place we cut. God has to do that." A dentist could pull a tooth out, but who's going to heal that socket? Medicine does not build tissue. Medicine only keeps the thing clean. A doctor might set a bone in your arm, but he can't produce the calcium and stuff that heals the arm. God has to do that. That's right.” [Faith Is The Substance, LA, CA, 51-0508]

“Medicine and doctors do not claim to heal. Mayo Brothers is one of our best authorities. They said, "We do not claim to heal"; said, "we only claim to assist nature. There's only one Healer; that's God." They can sew a place up in your hand, but they can't heal it. They can only sew it up. A dentist can pull a tooth, but who's going to heal the place? A doctor can take appendix out, but who's going to heal the place he cut? God does. Medicine doesn't grow tissue; medicine's only a aid. God is the Healer. A doctor can set your arm; but when he comes and sets your arm, and you don't go out and you're not well... The doctor does his part, what he's taught to do, and knowledge to do. He sets your arm, but he leaves it for God to heal. That's right. All healing is of God. So now, you just believe Him.” [The Deep Calleth To The Deep, Washington DC, 54-0624]

“Brother, when you see such motives as that... It seems to me, that I would be justified in saying this: it's selfish motive. Now, we know the surgeon helps; we know the medicine helps; we know chiropractic, osteopathic, we know all that helps. And after all, it takes God to do the healing, regardless of what comes or goes. God's the only One that can heal. There never was a drop of medicine that ever healed a man, or a mentally balanced man ever said it did. If he--if he did, I can just show you in a minute how--how foolish it is. See? There's only one Healer; that's God.” [The Great Commission, Edmonton, AB, 57-0804A]

“.I--I honor and believe in medicine and operations. I believe they're God-sent blessings to us. But sometimes we get to a place to where it's beyond what our medical science can--can handle. Then when it comes to that, I think, instead of giving up and dying, we have a right to come to the Great Physician.” [It Is I Be Not Afraid, Long Beach CA, 61-0213]

“Now, you say, "You believe in taking medicine?" I believe in anything that helps the human race is godly. Certainly it is. Exactly it is. Tell me why we have the--what would we do today without these clinics and hospitals and so forth? Certainly. I know there's some shut behind the door when I first started off, but where'd it all go to? See, you've got to look at anything sensible. Anything... If--if medicine, hospitals are not of God, they're antichrist. Burn them down, get them out of the country. Sure. But it's something to help you. Certainly” [Why, Beaumont TX, 61-0125]

“ Here not long ago, a certain sister that goes to this church, was at another church, where God was making everything happen. And this sister said to the other sisters, said, "You know, that man that could pray for the sick," said, "must have a--a very victorious life." And said, "He must just be able, his family and everything, be healed at a spoken word, like that." [Brother Branham snaps his fingers--Ed.] And the other lady happened to be from Jeffersonville. (And I'm sure that nobody has to know, but what Jesus said, "Among your own people..." And you--you know, your country. That's right. That's the reason I, it may be that right now coming close, it may be a change of time. You see?) And she said a--she said, "You know what?" Said, "Not one of his kids can have a sniffle 'less he takes it to a doctor." A poor, degraded, deluded thing like that. See? See, a woman that just wanted to throw off... Said, "When his children gets sick, he takes them to a doctor." [Taking Sides With Jesus, JEFF IN, COD, 62-0601 794-259]

“Anybody that's sensible will do the same thing. See? People can't understand that medicine is sent of God. Why, brethren, if it's not, it's of the devil. Sure it is. God is where medicine won't reach. Certainly. Medicine is of God, "Well," you say, "I know a lot of doctors that's..." Oh, yes, and I know a lot of preachers that's the same way too. It's not the man that's handling it, it's what it is. I know a many man's handled the Word of God don't believe in Divine healing, don't even believe in God. Right. But they handle It just the same. There's many a man out there with medicine and surgery and stuff that denies God and everything else, but there's a many one that believes Him too. If it helps people, it's of God. I don't have to--I don't have to take that car and ride home tonight; I can walk if I wanted to, but God made me a car, so I thank God for it. All these things come from God, but use them sensibly; don't go insane with them. See?” [Taking Sides With Jesus, JEFF IN, COD, 62-0601 794-260]
(We truly appreciate the help of Brother Donny Reagan in researching answers to the question above)