There is a pre-built tool called Search and Replace (recommended in the WordPress.org codex) that you can use to safely edit your database for just this type of change. Whether you choose to use a protocol relative link
//wp-content/... or change everything to
https to prevent mixed content is still your choice.
The reason for this tool, rather than the easier SQL replace is that data is
serialized in the WordPress database and if not taken care of properly can/will be destroyed.
Always make a backup of your database before using this. The toole also gives the option of running a dry run so you can see what tables will be affected before actually performing the task. I personally recommend a dry run (for obvious reasons) and then write down the tables affected. From here, if your server is not exceptionally fast it will be wise to select only a small amount of tables to make changes to at one time so that the process does not hang up and fail. Writing down the affected tables means you can skip the unaffected ones entirely in the live run.
It is not without risk, but if done properly you can run this on a live site. Since you will be changing valid, but not 100% accepted links into valid, and 100% accepted links it should not negatively affect your site so long as you do not give it so high of a workload that the process hangs. Also, mistakes on a live site are bad so don't make those.
If you have a construction area to use, then absolutely use that. But if not, I am saying that it is possible to run this tool on a live site without any trouble. It is by no means fool proof though. I use this tool a lot for moving sites from construction (etc), if this is your first time... expect trouble, always expect and plan for the worst. Make a backup and have it ready.
Note on version 2.? vs 3.?:
If your server is older, then version 2 is likely the one for you. If your server is newer then give version 3 a try. They are both fairly easy to use, but if your server accepts v3 then use that one, it's newer and does everything all one one page which makes things very much more convenient. Try a dry run with version 3 and you should find out if your server can run that version or if you need to use the older version 2.
As the tool should let you know, DO NOT forget to remove this once you are done using it. You don't want this left up for any nefarious character(s) to exploit.