Does Metro-North go to Clinton Correctional Facility?

No, it doesn't go anywhere near there. Clinton Correctional Facility is in Dannemora, New York, up near the Canadian border. Metro-North only goes as far north as Poughkeepsie and Wassaic, which are 200 miles south of Dannemora.

Neither Amtrak nor Greyhound buses stop in Dannemora, either. I think the best way to get there is to take Amtrak to Plattsburgh, NY (about 15 miles east of Dannemora) and hire a taxi to take you the rest of the way.

See the first Related Link below for the Amtrak homepage. In the "From" field, enter NYP for New York Penn Station (or, if you're not coming from New York City, use the Station Locator to find the station closest to you), and in the "To" field, enter PLB for Plattsburgh, NY.

See the second Related Link for taxi services in Plattsburgh. I called the first number on the list (City Taxi at 518-561-7777) and they said they would charge $26 each way ($52 round-trip) to take you to the prison.