TL;DR : Lock your credit accounts:
Basic recommendation, freeze your credit report. This does two things. First, it protects you. Second, if most US taxpayers freeze their credit it will change the way these companies do business. They're collecting information about you and reselling it to third parties. They'll charge you a fee (credit monitoring) to protect that information. In my opinion it is perverse that they'll only protect your information for a fee. It will be interesting to see how the class action law suits which follow will shape the credit monitoring services.
I'm also interested in how the Internal Revenue Service (IRS) of the United States addresses this. They've been pretending for two decades too long that the social security number is somehow a shared secret. It is not. Time to re-key, IRS. You have suffered a data breach through an irresponsible vendor partner. You've allowed these vendors to leverage your information for far too long. Fix this broken system.