What do you do if your social security number is on the dark web?

Monitor your credit reports and credit cards. You might want to freeze your credit. File your tax return as so as possible to prevent another from using your SSN. 

In addition to what Bsch4477 said, you should contact the IRS to get an Identity Protection Pin.  That way, nobody can file taxes using your Social Security Number except you.  You can read more about IP Pins and how to get one at this link.