Crimes of yesterday look much different than the crimes of today, or even tomorrow.
Crimes that were rare and effectively didn't exist just a few decades ago:
1. Computer fraud crimes
2. Identity theft crimes (among the fastest growing criminal enterprise globally)
3. Child pornography, sex trafficking, etc.
4. Hard drugs like crack cocaine, meth, bath salts, spice, etc.
The point is that nobody - you, me, etc. - can predict what criminal minds will concoct for fraud or money or whatever.
In the military we were trained that small bits of otherwise irrelevant information can be put together like a puzzle to create a big picture.
Given that ANYBODY in the world can access your gun and serial number based on a photo or video, I'd be very cautious posting any identifiable information online. That includes home address, social security, names of family members, number of family members, routines, etc. Once out there it lasts indefinitely.
For instance, I can go on Facebook and literally find the full names, addresses, phone numbers, vacation habits, hobbies, job/hours of work, common routes to work, their "check in" information, etc. of a lot of strangers on their public pages. A few more clicks and a little investigating and you can find a lot more information... if a nefarious person were targeting an individual it would not be hard to figure out everything important about them, and just watch until they post that they are going on vacation, for instance...
I don't know HOW serial numbers could be used nefariously, but can probably come up with a few possible ways.
I'd be cautious and carefully cover them up, and edit out any sensitive information you don't want permanently out there.
Let's just pick a very simple nefarious ploy. You post a video of XYZ wizbang gun with serial number obvious. Someone sees it, decides he wants it, and reports it as stolen. At the minimum, you'll get a visit from law enforcement, and a brief inconvenience of having to pull up receipts and proof. Gun may get confiscated temporarily while you prove ownership. Assuming you can, it may be a slight inconvenience. But what if you cannot satisfactorily proof ownership? Then you're looking at needing a lawyer, being on the defensive, having the gun put in an evidence locker, and then it's your word against the other party with neither of you having much more claim of proof... Then a prosecutor needs to determine who is telling the truth, and even if there's a 1% chance you lose, that is a BIG problem because firearm theft is a felony, and either way you'll be out $ thousands in legal fees.
And it's a risk-free crime for the other person, because it's difficult to prosecute unless one can prove the chain of ownership. So if they know it's a used gun, they can have a chance to score a new gun.
Perhaps I'm overly risk averse from my time in the military and as a lawyer, but I don't take unnecessary risks like this, even if they are very small.