Add a security layer to our SQL Servers (currently accesible from remote sql management studio)

We have a big system running with thousands of users (some from android apps, other from the web app, etc.).

The system is distributed, with databases in two locations (within the same country). In one location there are 5 servers in the same network, and each one has a copy of the database (via replication).

Among the software developers, a few have direct access to the production databases. Sometimes due to technical support requested by users to modify some operations not possible from the system itself, the developers/support team have to access the database directly and modify some records.

We know this is not the ideal manner of working. But it's been like this since years.

Recently we have found a few problems. One day one person updated hundreds of records from a table by mistake.

Since then we are analyzing how to improve this access.

We are looking for some way of improving the security. We would like to have a two-phase authentication system in place. Something that asks the user for two passwords when accessing from Sql Server Management Studio...

Is that possible? Or is there any other approach we can use to improve the security but still allow devs/support team to access the production database when necessary?

Users also (currenty) have access via remote desktop to all servers.

At least we would like to KNOW when this access is being done.