Is it safe to put the passport I'd on GoEuro site when you purchasing bus ticket

I assume you mean passport or ID card number. What is your objection to supplying your passport number?
What is the risk in doing so? Why should it not be safe? What is the difference between supplying a passport number and your name?
Sorry but I don't understand your reticence.

I can well understand your hesitation. If you are not familiar with GoEuro (and I sure am not), I would be very concerned about sharing passport into with an on-line provider without knowing that they are legit, safe, fully vetted and don't have a history of being hacked.

If someone lives outside Europe and has never purchased bus tickets before, a sense of hesitation is very wise. The site maybe totally safe and very reliable, but if you don't know, you just don't know. So, smart to ask.

I would be very concerned about sharing passport info

Maggie, I do not understand what the risk is in sharing passport info. What could be done with it that would be disadvantageous to you?

Passport number good in case something happens they can account for you.

Basically, if someone hacks your passport number, there is no way to "steal" your identity or otherwise cause you harm. It is not a credit card number, which I assume you are also giving them in order to buy tickets. That would be a problem, but not much. I've been hacked twice, once by a dishonest pizza employee in Colorado. Both times cost me nothing and had a new card fedexed to me immediately upon discovery.

The passport is a travel document, so it has no consequence whatsoever if someone has that number.

Ditto! :))

For out confused European posters, this is a societal difference. Here, passports are only used for travel. We are not required to give them over to hotels for registration, nor used as identification on a regular basis. Chances are people have to go to bank safety deposit boxes to get them before a flight. We have it drilled into our head to keep them safe and secure. There is also the constant media of identity theft (thus the high sales of RFID protection).
Over time, one equates the actual physical passport protection to include the number as well.

... Maria, neither do us here. Passports are only for travel, nothing else. For anything else, we have our National IDs -except UK and a few other.

Maria, passports are travel documents everywhere. When you are in another country, it's the only valid form of identification-, as it is official.
In Europe, and anywhere, locals don't go around with their passports, and it is not a form of identification in the person's own country.

I misunderstood; I thought the original poster was saying they would have to upload a copy of his/her entire passport, since I read "put the passport". If it is just the passport number, then I would have no problem with that.

Likely the link the original poster provided would have told me that, but I do not click on links that are posted (for security reasons), if I am not already familiar with the site and know it is the correct address.

So, if it is just the passport number needed, then I would not have an issue IF the vendor that runs the site is otherwise legit and well vetted.

A passport number has no meaning or value to anyone. Not like a credit card number. Don't get overly paranoid. If you go to Italy and check into a hotel they will make a photo copy of your passport.