The question behind this one is usually financial rather than practical: if I spend money fixing the screen, do I get more than I spent back? For a phone you are selling, the answer is almost always no, and it is worth understanding why rather than taking that on trust.
There is one situation where repairing first is the right answer, and it is at the end.
Why the arithmetic does not work
A buyback offer on a damaged phone is roughly the working value minus what it costs us to make it working. We repair at trade rates, in volume, with parts bought in bulk. You repair at retail, one panel at a time.
So when you pay retail to fix a screen, you are spending more than the amount the offer goes up by. The gap is the repairer's margin, and you are paying it for the privilege of increasing our offer by less than you spent. On a Samsung specifically this gap is wide, because AMOLED panels are among the more expensive displays in the industry.
This holds at almost every price point. The higher the model, the bigger the numbers on both sides, but the direction is the same.
The green line is not a crack, and it is not your fault
A thin vertical green line down a Samsung display is a known panel failure rather than impact damage. It tends to appear spontaneously, often after a software update, on phones that have never been dropped. It is common enough that we see several a month.
It is priced as a panel replacement, which is a substantial deduction, but it does not disqualify the phone and it is worth declaring accurately. Describing it as a cracked screen would actually understate what is wrong, since a crack sometimes leaves the display working perfectly.
Burn-in is a third thing again
Years of the same navigation bar, the same status icons or an always-on display can leave a faint permanent ghost on an AMOLED. It is most visible on a plain white background in good light, which is exactly the test worth doing before you describe your phone's condition.
It is extremely common on any Galaxy past its third birthday and it is not a reason we would decline a phone. It is a panel condition issue, so it lowers the offer, and it lowers it by less than a crack does.
What about a cracked back?
Much less serious than people fear. Rear glass is considerably cheaper to replace than the display assembly, so the deduction is smaller, and on some models the back is a purely cosmetic repair rather than a functional one.
The exception is a phone that was dropped hard enough to crack both sides, which usually means the frame took a knock too. Mention that separately if it applies.
The one case where fixing it first is right
If you are going to keep using the phone for another six months, repair it. That is not a resale decision, it is a usability decision, and the money buys you six months of a phone you can actually read.
What does not make sense is repairing a phone specifically in order to sell it. If it is going anyway, send it as it is and let the offer account for the damage.
The short version
Repairing to sell loses money, because you pay retail for a repair that raises the offer by less than it cost. A green line is a known panel failure rather than damage you caused, burn-in is normal on an older AMOLED, and a cracked back is cheaper than everybody assumes. Repair only if you are keeping the phone.