Are you talking about a country besides the US? And you're talking about a commercial bank, right? In the US, banks don't buy gold from consumers. The last time they sort of did (in the early 1900s), they were trading gold coins for gold certificates, and then they later stopped allowing consumers to trade them back. This is known by a well-known financial term: "Gotcha, suckers!"
If someone were naive enough to deposit a $50 Gold American Eagle today in a bank, the depositor will get a credit of $50 on their account, and later some clever person will ask the teller if they have any "strange money" lying around, and that lucky person will be able to withdraw a $1,700 coin for $50, if it lasted for even a second in the teller's drawer.
But let's say you're going to a place that does indeed still buy gold coins.
The discount depends on the type of coin, and the type of damage. An old (collectible) coin has a part of its value set by the gold value, and part by the collector's premium. Better specimens command better collector's premiums, so a damaged coin, as long as it isn't a chunk of the coin missing, won't be worth less than the melt value. (You may not get that much from a dealer, but it should be fairly close.) If part of the coin is missing, then the person buying it should weigh the coin and adjust the price proportionately.
It's likely, though, that if you have the items in a safe, you may have a puddle or blob of gold, but it should still all be there unless someone takes it. Gold melts at about 1850 degrees Fahrenheit, but it would take half the surface temperature of the sun for it to boil away. If it's unidentifiable, it may need to be assayed again.