These answers didn't help me out. I really didn't need any crazy scripts.
I had created a public key on my client machine in git bash and was trying to copy it to a VPS.

After creating your public key, the key should be stored as "(whatever folder you started in)/.ssh/id_rsa.pub"

So use this command:
cat ~/.ssh/id_rsa.pub | ssh user@123.45.67.89 "cat >> ~/.ssh/authorized_keys" where user is your username (sometimes "root", or whatever you may have set up), and replace 123.45.67.89 with your machine / host / VPS's IP address.

If the directory .ssh is not yet created on the host machine, use this small variation:
cat ~/.ssh/id_rsa.pub | ssh user@123.45.56.78 "mkdir ~/.ssh; cat >> ~/.ssh/authorized_keys"

ssh-copy-id does a couple of things (read the man page for details), but the most important thing it does is append the contents of your local public key file to a remote file called authorized_keys.

• You could do this yourself by opening the key file with a text editor and pasting the contents in the Kitty terminal.
  echo 'long_line_with_contents_of_public_key_file' >> .ssh/authorized_keys




• Alternatively, you could upload the file using WinSCP (which uses sftp, or scp as a fallback) and do something similar to my previous suggestion, without the ugly copy/pasting.
  cat id_rsa.pub >> .ssh/authorized_keys
  
  where id_rsa.pub is the filename of the public key you uploaded.

Inspired by zoredache's answer, I've created a bunch of scripts that are the windows version. However they all depend on plink.
Please take a look here

https://github.com/VijayS1/Scripts/blob/master/ssh-copy-id/

I also have a winscp script that can be used as per another answer. :)
Excerpt from the readme:

Attempted methods so far:

• DOS(.cmd) - Success
  
  
  
  
  • usage: .Scriptname test@example.com password [identity file]
  
  
  
  


• VBS (.vbs) - Success
  
  
  
  
  • usage: .Scriptname /i:idtest.pub user@example.com /p:password
  
  
  
  


• Powershell(.ps1) - Success
  
  
  
  
  • 
    usage: .Scriptname -i idtest.pub user@example.com password
  
  
  
  


• mremoteNG (ext app) - Success
  
  
  
  
  • Select Host, right click, external tools, select Scriptname
  
  
  
  


• WinSCP script (.bat) - Success
  
  
  
  
  • 
    # "WinSCP.com" /script=".Scriptname" /parameter "user[:password]@example.com" "id_rsa.pub" [/log=".copyssh.log]"
  
  
  
  

In Windows 7 there is a ssh.exe

Here is what worked for me:

1. create identity (on windows)

c:>ssh-keygen

That created an identity file in the home directory.
I changed the name of the public key to "id_rsa"

2. copy the file to the target linux system using the ssh
Credits to https://serverfault.com/users/984/zoredache for his answer

c:>ssh user@lnxhost "umask 077; test -d .ssh || mkdir .ssh ; cat >> .ssh/authorized_keys || exit 1" < \path_to_where_the_file_was_generated_from_ssh_key_genid_rsa.pub

Note: For some reason piping didn't work for me:

# this should work but it didn't work for me 

type file | ssh user@lnxhost "cat >> /tmp/t.txt"

3. Correct the file on linux
The id_rsa.pub file on windows is multiline where linux expects it in in a single line so we have to correct it a bit. Login to linux and open the file:

vi ~/.ssh/authorized_keys

For example:

---- BEGIN SSH2 PUBLIC KEY ----

Comment: "2048-bit RSA, user@winhost"

AAAAB3NzaC1yc2EAAAABIwAAAQEAnvYlVooXGoj3+7huZBUqf4wj57r25SHCKiiShyla33

5flX7Rsmb4meExpdh2NzfzffG15xl1wo0xBZ3HdZdqF2GUniEcNbtVjS1FKzQwPfsYPHMC

Y58qT0U2ZgK1zsXj2o0D2RWrCv3DFFfwUgNyZRYN2HK32umY6OmGSOVuJvIKhT+X6YaCVy

ax3CHv2ByB2OTBl9mh4nrwYAVXToT+X2psBE+MKB5R85lrUGkl3GtymTk10Dvf5O80exdT

LFRMvkCA5RAIZgvxMk/bbNaH/0UHQoctX9oaDeKGWUPfVaknFBQdU9009+lK/ocAlKVNHE

Qkw+1wuV6dFoT1/hngSw==

---- END SSH2 PUBLIC KEY ----

should become

ssh-rsa AAAAB3NzaC1yc2EAAAABIwAAAQEAnvYlVooXGoj3+7huZBUqf4wj57r25SHCKiiShyla335flX7Rsmb4meExpdh2NzfzffG15xl1wo0xBZ3HdZdqF2GUniEcNbtVjS1FKzQwPfsYPHMCY58qT0U2ZgK1zsXj2o0D2RWrCv3DFFfwUgNyZRYN2HK32umY6OmGSOVuJvIKhT+X6YaCVyax3CHv2ByB2OTBl9mh4nrwYAVXToT+X2psBE+MKB5R85lrUGkl3GtymTk10Dvf5O80exdTLFRMvkCA5RAIZgvxMk/bbNaH/0UHQoctX9oaDeKGWUPfVaknFBQdU9009+lK/ocAlKVNHEQkw+1wuV6dFoT1/hngSw== user@winhost

4. test it

c:>ssh user@lnxhost "ls -al /tmp/"

This should list the content of /tmp without asking for the password.

If you do not have ssh-copy-id on Windows, you can run it on the server itself.

• In PuTTYgen, load your private key (.ppk);


• Copy the contents of the box Public key for pasting into OpenSSH authorized_keys file to a clipboard.


• Paste it into your favorite editor (Windows Notepad will do).


• Save the contents to a file with .pub extension.


• Upload the .pub file to the server.


• Login to the server with an SSH client, like PuTTY.


• 
  

  On the server type:

  
  
  

  ssh-copy-id -i mykey.pub username@localhost
  
  

  
  

On Windows ssh-copy-id script comes with Git for Windows. So you may use that locally, if you have Git for Windows.

If you do not want to do this manually, you can use WinSCP 5.15. It can setup the public key authentication for you.

Use Tools > Install Public Key into Server button on SSH > Authentication page of WinSCP Advanced Site Settings dialog.

(I'm the author of WinSCP)

If you are using cmder (or msysgit/mingw that has scp & ssh), I just wrote a simple python script for this. It can be found here: https://gist.github.com/ceilfors/fb6908dc8ac96e8fc983

Sample usage: python ssh-copy-id.py user@remote-machine.

Password will be prompted upon running the script.

what i did, having CygWin on my Win10, connecting to Linux (based on aboves answer):

- note: using cat, it would resolve the cygwin path automatically, as well as any cygwin command using the cygwin-linux-folder-structure

1. added c:cygwinbin to the environment's Path variable

2. starting cmd.exe (windows commandline)

3. > ssh-keygen -t rsa   (just pressing enter till done)

4. > cat ~/.ssh/id_rsa.pub | ssh user@server "umask 077; test -d ~/.ssh || mkdir ~/.ssh ; cat >> ~/.ssh/authorized_keys"

5. ..enter server password

6. > ssh user@server (testing, not beeing asked for password)

This following steps would do:

STEP-1: Generate RSA Key Pair

C:Usersuser>ssh-keygen

Generating public/private rsa key pair.

Enter file in which to save the key (/c/Users/user//.ssh/id_rsa):

Enter passphrase (empty for no passphrase):

Enter same passphrase again:

Your identification has been saved in /c/Users/user//.ssh/id_rsa.

Your public key has been saved in /c/Users/user//.ssh/id_rsa.pub.

The key fingerprint is:

20:16:9b:0d:00:92:c4:34:99:51:20:b7:ef:50:c4:0f user@localhost

STE2-2: ssh-copy-id equivalent in windows

C:Usersuser>ssh user@remote "umask 077; test -d .ssh || mkdir .ssh ; cat >> .s

sh/authorized_keys || exit 1" < "umask 077; test -d .ssh || mkdir .ssh ; cat >> .ssh/authorized_keys

 || exit 1" < C:Usersuser.sshid_rsa.pub

The authenticity of host 'remote (xx.xx.xxx.xx)' can't be established.

RSA key fingerprint is 99:99:73:74:fe:14:bc:91:c8:3b:ac:f4:95:99:4d:06.

Are you sure you want to continue connecting (yes/no)? yes

Warning: Permanently added 'remote,xx.xx.xxx.xx' (RSA) to the list of known hosts.

*************************************************************************

This computer system is the property of Sample Corporation and is to

be used for business purposes.  All information, messages, software and

hardware created, stored, accessed, received, or used by you through

this system is considered to be the sole property of Sample Corporation

and can and may be monitored, reviewed, and retained at any time.  You

should have no expectation that any such information, messages or

material will be private.  By accessing and using this computer, you

acknowledge and consent to such monitoring and information retrieval.

By accessing and using this computer, you also agree to comply with

all of Sample Company's policies and standards.

*************************************************************************

user@remote's password:[Enter Password for first time]

STEP-3: Passwordless authentication works!

C:Usersuser>ssh user@remote

*************************************************************************

This computer system is the property of Sample Corporation and is to

be used for business purposes.  All information, messages, software and

hardware created, stored, accessed, received, or used by you through

this system is considered to be the sole property of Sample Corporation

and can and may be monitored, reviewed, and retained at any time.  You

should have no expectation that any such information, messages or

material will be private.  By accessing and using this computer, you

acknowledge and consent to such monitoring and information retrieval.

By accessing and using this computer, you also agree to comply with

all of Sample Company's policies and standards.

*************************************************************************

Last login: Wed Oct 14 14:37:13 2015 from localhost

There is a Windows version of ssh-copy-id I found on GitHub:
https://github.com/zhengyi-yang/ssh-copy-id/tree/master/dist

Powershell version for SSH included in Git For Windows

In fact, it can work as long as you have ssh in your path.
Add the following in your powershell profile:

function ssh-copy-id([string]$userAtMachine){   

    $publicKey = "$ENV:USERPROFILE" + "/.ssh/id_rsa.pub"

    if (!(Test-Path "$publicKey")){

        Write-Error "ERROR: failed to open ID file '$publicKey': No such file"            

    }

    else {

        & cat "$publicKey" | ssh $userAtMachine "umask 077; test -d .ssh || mkdir .ssh ; cat >> .ssh/authorized_keys || exit 1"      

    }

}

In a powershell console:

ssh-copy-id user@machine