Recently our group of friends has plunged into D&D with me as DM. After the first session, one of the PCs came to me with a good question on balance between constantly perception checking everything and rushing forward. Is there any advice you can offer?

To elaborate, strategically, it always seems best to constantly check everything. Every new environment, every new person, every object. Best by everyone, so that you have a better chance that it will yield something. With these checks flying about, the PCs made really slow progress. But by never doing checks, you of course make mistakes and get hurt.

The balance seems a bit hard to find, and the PC wondered whether there is like a "normal" amount of checking?

Are there ways, as a DM to encourage people that do not constantly want to check everything? Or should I approach it differently?

This is where passive perception comes in. It should be an entry on the character sheets here, but it basically amounts to 10+ perception score. This represents stuff a character will just happen to notice without really specifically looking for it. Have the players tell you everyone's passive perception, and then just as you go along, compare that to the DCs of things that might otherwise be hidden from them.

Otherwise, this is just something new players tend to do. Give it some time, and you'll find it tapers off, generally, after they realize that a Trap of Certain Death isn't waiting behind every corner. maybe every 3rd or 4th corner, but not every corner.

Well that's definitely true, and I hope this will stabilise indeed in the future. Nevertheless (we're doing Lost mine of Phandelver btw), the mechanics seem to favour the overly cautious, despite the slow pace.

It's for this that I was wondering whether I could help them realise what times are good to actively perception check or insight check, and when to not bother. But perhaps this might be something to just relay to them in between sessions then?

The other thing is not not let everyone make a check. One character searches the room, one character make insight against the new person. Either whoever says it, or let the choose amongst themselves. 

And, really, only you should call for a check, not them. Checks are for when there is a chance of success or failure. They say they search the room, and you can tell them what they found. Sometimes there’s nothing to find beyond what you described, so there’s no need for a roll. Now, sometimes you can have them roll even when there’s nothing, just to keep them on their toes, but the risk there is they get a 25 and you say there’s nothing, it can get really suspicious. 

Quote from Xalthu >>

The other thing is not not let everyone make a check. One character searches the room, one character make insight against the new person. Either whoever says it, or let the choose amongst themselves. 

And, really, only you should call for a check, not them. Checks are for when there is a chance of success or failure. They say they search the room, and you can tell them what they found. Sometimes there’s nothing to find beyond what you described, so there’s no need for a roll. Now, sometimes you can have them roll even when there’s nothing, just to keep them on their toes, but the risk there is they get a 25 and you say there’s nothing, it can get really suspicious. 

 This is what I noticed happens on Critical Role. One person makes a check, on behalf of the group. Maybe another person gives them advantage, but there's only one role. And that's it.

That's definitely a good idea, although even of CritRole, there are rare occasions where it is a vast room and the DM asks everyone that joins in the search to role.

But point taken indeed!

Any other tips that could be used to "punish" the over-cautious player? I mean, the Leeroy Jenkins types typically understand the danger behind what they're doing, but I don't see (for the players) enough downside to play it too cautious and to suspect everything and everyone (apart from the overall gale momentum and enjoyment of course). It would be great to be able to provide a downside to trying to check every element from the multiverse...

The downside of constant checking IS the slowing down of the game, IMO. As the DM, you could drop more hints about things that might require a perception check, and what might not. For example, "you walk into the dungeon's Stony room, and everything feels just a little bit too quiet. Like the room itself is holding it's breath" - players would probably want to roll perception / investigation to figure out what's going on.

That, plus the fact that rolling a perception check on an empty/safe room shouldn't really do much. "I want to check around the room. Nat 20! What do I find out?" ...."It's empty." Or "you don't find anything beyond what I've told you" should suffice. You don't need to make things up just to support a high or low roll, sometimes a room really is just what you see on the surface level. 

Last bit: passive perception is great, but I use it a bit different from most people I think. I'll keep a tab of the top two ish people with highest passive perception, and ask THEM to roll a perception check if nobody is taking my hints about an area. 

That, plus the fact that rolling a perception check on an empty/safe room shouldn't really do much. "I want to check around the room. Nat 20! What do I find out?" ...."It's empty." Or "you don't find anything beyond what I've told you" should suffice. You don't need to make things up just to support a high or low roll, sometimes a room really is just what you see on the surface level.

 Yeah if the room is empty and they roll a Nat-20 then all that means is their character is absolutely certain that the room is definitely empty