My personal preference is to keep the most recent backup, the one from the first backup of the current month, and one from the first backup of the current year. So at all times I have 3 backups. Disk space is cheap, and having three backups of my 1GB library isn't a big deal to me. Everything else I trash.

You could just keep the most recent backup, but I have heard of people having a corrupt backup, even though Lightroom checks the integrity of the file before a backup. Personally, I think having more than one backup is just fine, since the tradeoff isn't significantly detrimental to me.

The principle product manager at Adobe - Jack Nack recommends keeping 2-3 library backups here. One opinion isn't the end all especially when it comes to backup strategies, but I think he offers solid advice.

Actually deleting these old backup files isn't going to do anything negative to your current library. Your current working library file has everything that you need for ongoing work. The backups are just that, stand alone files that can be used for recovery purposes in case of corruption or other emergencies. Keep in mind, the older the backup, the less fun it will be to use it in the case of a recovery. But I also believe the older ones may be more stable to some degree.