There are two easy options.
dig deep, mobs spawn like crazy down there, and they are harder then the mobs on the surface = better loot find the corruption and park yourself on the edge of it, you will be attacked by soul eaters, super easy to kill and they drop silver coins most of the time
gaming.stackexchange.com/users/2771/Šimon Tóth
Although not the fastest way to "farm" silver coins, as such - could you just invite a friend who already has 50 silver coins into your world as a multi-player environment, and then the merchant will spawn assuming you have the right house for him?A world made available on multi-player will still be available for you as a single player afterwards.
As Camster points out below, they would have to have 100 coins, and give 50 to you - as each player needs to have 50 coins before he will spawn.
gaming.stackexchange.com/users/4872/jos
Since you won't have access to a merchant, the best way would be to get some decent equipment (i.e. a copper sword, though wood would also be fine) and go spelunking.
Get as many pots as you can.
I personally just got lucky and found a pinky slime on my first day - 1 gold coin.
gaming.stackexchange.com/users/7111/The Communist Duck
If you're willing to risk going into the corruption, harvest two vile mushrooms.
Turn those vile mushrooms into vile powder and sprinkle rabbits and goldfish with it.
The resulting corrupt bunnies and goldfish are easy to kill one at a time and drop six and five silver coins respectively.
After at most ten fights you'll have enough cash to attract the merchant.
gaming.stackexchange.com/users/7992/Fambida
I managed to get a lot of silver coins by killing the Jungle slime, they drop about 5 silver coins each time and they are simple to kill.
Also don't die if your on normal mode as you will lose your precious silver coins...
a good way is by using a chest and keeping all of your coins in there.
You could also turn 100 bronze coins into 1 silver coin if you use a work table.
gaming.stackexchange.com/users/169807/Abby
This is a little dangerous, but if you can summon a meteorite, the Meteor Heads are easy to hold back, will be the only thing that spawn while in the area, and drop a good amount of money.
They are probably the easiest flying enemies as well due to their fairly straight pathing.
Hope this helps.
gaming.stackexchange.com/users/8581/James
The easiest way to get the silver coins is by killing demon eyes.
They drop 1-2 silver coins each.
Take a strong sword (silver sword will do) and
some armor then slaughter a lot of them.
gaming.stackexchange.com/users/70617/terraria pro
There are 4 basic ways to get money: Killing Enemies/Bosses, pots/chests, angler quest rewards and selling items to NPCs Since the 1.3 update there has been an Expert Mode that will reward additional loot from chests, money dropped from enemies and better chance of rare drops.
This will help get money early game.
Be Warned!
While expert mode does have it's perks the enemies are much harder with an increased 2x Damage and 2x Health and those buggers can make off with your hard earned cash!
If you can, locale the thief with their sparkly appearance meaning they have stolen coins of the colour they sparkle.
You also drop 75% of money (for software characters)
so it is always a good idea to have a chest or piggy bank to store money.
Pots have a chance of spawning a coin portal that gives you 5-15 gold so another good strategy is to run across the surface and smash all the pots you can find.
gaming.stackexchange.com/users/74117/Oreo
I am trying to attract the Merchant on my world and according to the  wiki , I need at least 50 silver coins.  What is the fastest way to farm silver coins in a new world?
How can I gain silver coins quickly?
gaming.stackexchange.com
<terraria>
gaming.stackexchange.com/questions/22940
gaming.stackexchange.com/users/730/Drake
Dig deep or go spelunking. You can also kill the jungle slime to get 5 silver coins each time and sell items to NPCs. Killing Enemies is the most prominent way of gaining silver coins rather quickly.
There seems to be a variety of ways to gain silver coins most of them having to do with killing enemies.