Starting to log crash dumps are a good start, but it might not tell you the whole story. Your users have a diversity of hardware/network connections and may use the game in ways you may not have imagined. The crash dumps may tell you WHAT the error was, but not what the conditions on the clients or even your server were at the time the crash dump occur.
To help you establish a full picture, you'll want to make use of logging and log not only crash dumps from your game server software but also server environment variables and things on your client related to how it interacts with your server.
From my experience running games, you don't want to get involved in creating your own logging tools. Log data can swell up into the giga or terabyte level, and between managing that volume of data AND building a tools, you'll end up using a lot of your server guys simply to maintain logging. It's a lot quicker to just use a 3rd party logging service like Loggly to get up and running.
Regardless of the path you pick to collect, here are the logs you’ll want to consider collecting:
Game server application logs: You can log most any information from your application to file or SYSLOG socket. So within your application code, log any crash or exception information. Also think about logging your gameplay mechanics for things that would indicate when they're failing so that you get a list of events that led to the error.
General server logs: Most everything from the server, system logs, web server logs, database logs, etc. This data can help you spot any abnormalities in server performance which might indicate why a bug is happening.
Client logs: A lot of online game bugs stem from high latency caused by a slow client-side network or from bugs in the client sending incorrect data or commands to the server. You want to log anything within the client that has to do with communication with the server or manipulation of user's data to catch these client-driven bugs.
As far as how you'd collect the log data itself
Try to avoid storing logs directly on live servers, logs blow up to huge volumes, and you really don't want to be managing log retention & purge issues on live servers. For anything you want to log you can choose to write it to file and then have a SYSLOG agent or send it directly to a SYSLOG socket and have it sent out to a provider like Loggly directly. For logging anything on the client, you'd simply store any data you'd want to log in JSON format and send it direct from the client using HTTP methods.
It sounds like you don't already have logging implemented, so you'll need to implement it into your server software and take it live. It would be elegant to somehow push this update while your game is live such it doesn't interrupt player experience, but if you don't have a way of doing this, I wouldn't spend a lot of effort on creating it just yet; one problem at a time. Just take down the servers at a time you know there are a low volume of players and patch logging-enabled updates then.