One of my players asked me during combat with archer enemy, if he can cut the bowstring of the bow making the archer basicaly powerless. 

Would you allow something like this? What about damaging any other weapons if they are held by enemies? How to handle it?

It would be extremely difficult, but possible. Let them make an attack roll against the weapon, but make it very difficult.

In general attacking carried items isn't possible and may not be a precedent you want to set, as it can make for a very annoying game and tends to be worse for the PCs (who have a budget) than the NPCs (who do not) -- sunder existed in 3.x, and got removed for a reason. If you decide to allow it, probably allow the wielder to make a Dexterity save (as if against a grapple) to prevent, and remember that many enemies have more than one weapon (for example, a scout has a shortsword).

I’d not allow it. This edition doesn’t have called shots — if you’re going to let them target a bowstring, why not aim for the head or eyes or hand? And there’s no real stats for equipment damage. Heck only a very few abilities allow you to disarm someone  This is a step beyond, to my mind.

Now if they wanted to sneak in the night before and sabotage the enemy gear, that would be fun. But once combat starts, no. 

Quote from Aasimar11 >>

One of my players asked me during combat with archer enemy, if he can cut the bowstring of the bow making the archer basicaly powerless. 

Would you allow something like this? What about damaging any other weapons if they are held by enemies? How to handle it?

 I don't allow this, simply because combatants protect their weapons and so hitting wielded object in combat is usually unlikely to happen both to PC, monsters or NPC, hitting the creature instead.

Metagamingly speaking, i would say it's preferrable not to go there as if the DM starts breaking PC weapons in return, things will rapidly become not cool.

Personally, I'd treat it the same as an attempt to Disarm the enemy, they can flavour it however they want - including cutting the bow string. But just remind the players that the enemies can also do this to them. 

From the DMG (optional) Disarm Action:

A creature can use a weapon attack to knock a weapon or another item from a target's grasp. The attacker makes an attack roll contested by the target's Strength (Athletics) check or Dexterity (Acrobatics) check. If the attacker wins the contest, the attack causes no damage or other ill effect, but the defender drops the item.

The attacker has disadvantage on its attack roll if the target is holding the item with two or more hands. The target has advantage on its ability check if it is larger than the attacking creature, or disadvantage if it is smaller.

To me this almost sounds like 'called shots' where a player will say, "I want to aim at his eye and try and blind him".

I don't allow 'called shots' in my game, and when someone asks, I just remind them that anything the players can do, the monsters can do too, and that is usually the end of the conversation.