Franklin County was established in 1811, but there are records going back farther when this region was the northern part of Hampshire County (possibly back to 1662).
If you are doing research on Franklin County public records that that time period, you will have to contact the county clerk's office in Greenfield. They have their own collection of records up until 1841 when the state took over the task. All you need to do is submit a request form and pay the proper fee in order to get a copy of any vital records.
The same goes for getting records between 1921 and the present, except you have to send the form to the Registry of Vital Records and Statistics (they're located in Dorchester) rather than your local office. You can use the same form though, available from the website of the Office of Health and Human Services for Massachusetts.
But for any Franklin County public records that fall between 1841 and 1921, you won't be able to just make a request. You have to do a bit more of your own research at the State Archives in Boston. They will do some look-ups for people who are not able to come in person, but they are not equipped to find things the same way the county office is. Its better if you search through their indexes yourself to find what you need.
When making requests for any type of Franklin County public records, the staff will be able to search for a 10 year period if you are unsure of the exact date of the event.
If you don't necessarily need an official copy of any of these records, you can also find many of them digitized and available online. Large genealogy websites often have large collections of these documents because they are not held private like in other states.
Franklin County Birth Records
The fee to get a certified copy of a Franklin County birth record is $28 if you make your application by mail, but only $18 if you go to the office in person. If you are searching at the State Archives, there is no cost at all.
Franklin County Death Records
Franklin County death records are going to cost you the same as a birth record will. If you send your request by mail, it will take about a month to get processed and returned to you. An in-person application is typically processed while you wait.
Franklin County Marriage Records
You can also order Franklin County marriage records for anyone in the county without restriction. The cost for certified copies is also $28 or $18 each.
Franklin County Criminal Records
Criminal records are not handled the same way as other vital records, and you will have to place your request with the Department of Criminal Justice Information Services rather than at a county office. These files have a fee of $30 for each one, and you are permitted to access your own Franklin County criminal record as well as any other record that has been released to the public (some are considered confidential).