Your "Broken" pickaxe is just a regular Gold Pickaxe, with a prefix. It is (in every way) still functional. It just has a modifier called "Broken".
According to the Terraria Wiki, the Broken modifier does the following:
This means that swinging your pickaxe will deal 30% less damage and 20% less knockback to enemies, when contrasted with a non-prefixed Gold Pickaxe.
As for "fixing" your Pickaxe (even though it theorically isn't fixing) - There are only two possible solutions to that.
- Craft another Gold Pickaxe.
Modifiers are applied randomly when a new tool is made. There is also a chance that it has no modifiers. It was just that when you crafted your initial Gold Pickaxe, you were unlucky to get that modifier.
- Reforge it (using coins) by paying the Goblin Tinkerer to do so.
Although you'll be guarenteed to be given another prefix/modifier, it may not be a positive one.