"How I should respond to..." depends on what kind of game you want to run
Every DM has their own style. Some like their games to be realistic, some like to strictly follow Rules as Written, some are avid believers in the Rule of Cool, and some don't really care about the rules at all. But whichever of these styles you play, that is probably the style that you enjoy, and since the point of the game is for everyone, DM included, to have fun, then you should (probably) respond according to whatever style it is you play the game.
Now, "play according to your style" sounds nice, but that doesn't actually tell you much, does it? So, I'll use your example to try to illustrate a few different styles.
Let's start with RAW
As Quadratic Wizard and NathanS pointed out in their answers, there aren't actually any rules that (directly) support this action. There simply are no RAW mechanics for catching a sword like this. But, as Quadratic Wizard also pointed out, there are quite a few rules you could easily use as a stand-in, and then follow NathanS's advice and just narrate the action of actually catching the blade with your hands.
Some rules that would allow you to do this are (again, pulled from Quadratic Wizard's answer) the Blademaster's Parry, the action-consuming Dodge, or readying a Grapple. However, as both Quadratic Wizard and NathanS have noted, you still have to narrate this actions as if the sword were caught.
Rule of Cool
The Rule of Cool can be one of the most entertaining parts of a game. The rule is, if it's cool, then let it happen. It's not an official rule in any sense, but I know quite a few DMs who nevertheless use this "rule" as a core guiding influence in their games. It makes it easy for players to shine and have their moments in the spotlight, and it's a pretty good way to make players feel like their characters really are awesome adventurers above and beyond the average commoner. Following this, a low dexterity (acrobatics) check of DC ten or fifteen might be appropriate for attempting to catch a sword.
This doesn't usually work well for very serious, realistic, or gritty games though, because it tends to break the tension and immersion. So if ridiculous and over-the-top awesomeness doesn't fit the style you're running, then this is probably not the solution to go with.
Here's how I would rule it
My own style is that I try to follow the rules as much as I can (in part because I am still trying to learn them), but I am willing to bend those rules whenever and wherever I think that I can make it more fun. So with that in mind, I would allow the attempt to be made.
That said, it is almost impossible to catch a blade swung at you. Your strength, reflexes, and luck have to be phenomenal to even have a chance! So with that in mind, I would set the DC to 30. 30 is right up there on the edge of what a high level PC might be able to accomplish, if the appropriate skill is strong enough and they get lucky, so that seems like an appropriate difficulty to me. Additionally, since this is a reaction to an incoming attack, this will eat up the character's reaction for that round.
Now, to actually catch the blade, an individual has to be strong enough to completely stop a blade mid-swing using only the friction between their hands and the blade! That individual also has to be fast enough to catch the blade mid-swing, with their hands closing on the blade. Close your hands too soon, and they get sliced! Close them too late, and... well, you get a sword straight to the face.
I don't like making two checks to perform one action, even though both strength and dexterity apply here, so I won't do that. Instead, I would tell the player to make either a Strength(Athletics) check or a Dexterity(Acrobatics) check to catch the blade. This does benefit the player a little bit, especially since I'm calling for a DC 30 check.
Once the check has been made, there will be consequences. I want those consequences to be interesting on both failure and success, so the player feels like their choice actually did something, so I'm not going to leave this is just a narrative fluff.
So, on a success, I would rule that the player actually manages to catch the blade mid-swing! The shock and surprise on the attacker is so great that all sources gain advantage against that target until its next turn! This grants a clear and powerful gain from a very difficult maneuver, rewarding the player if they can actually pull it off!
On the other hand, failing to catch the blade means that you straight up take a sword to the face, with no attempt to dodge, so... if you fail to actually catch the blade, then the hit is an automatic crit. This also means that an attack that your AC would have ignored becomes a hit because you tried to catch the blade. This is a bit harsh, I'll admit, but the character is literally just taking a sword to the face without all of the abstracted blocking and dodging that normally goes into landing a hit.
Of course, the most important part of this would be for me to tell the player a.) that this will be an incredibly difficult check, and b.) exactly what the consequences are for success or failure. This way, I'm not blindsiding them with a crit out of nowhere, and the player can make this decision with full knowledge of both risk and reward - an essential part of making an informed decision.
Now... a DC 30 is very high, and would exclude all but the strongest or most dexterous of characters. If I decide that I want this to be an option more often, then I think I'd make a dynamic difficulty check. An easy way to do this might be a difficulty check of the enemy's modified attack roll + 10 (so, a roll of 12 with a +5 to hit mod (that's a 17 versus AC) becomes a DC 27 Athletics or Acrobatics check to catch the blade).
Of course, if you're going for strict realism, then you have the easiest answer of "No, it's impossible." While this might be a little bit less fun for the player, it is a reasonable, and easily defended, ruling that doesn't require either further justification or any remotely taxing decision-making. DM-ing can be very taxing, and every decision drains your creative well just a little bit. The more complex the decision is, the more draining it can be, so it is perfectly reasonable to look at a (near) impossible task and just say "No, your character is not able to do that."
There is no particular way you should respond
The TLDR of all this is that everyone has their own style, and everyone thinks different things are fun. Choose an approach that fits your game and your style, make it fair, and stick with it!