In case you're having this problem in Windows, keep in mind that the public/private keys that you might use to connect to a remote machine from WSL aren't the same ones that VS Code will use to connect from Windows. You need to create a separate public/private key pair for Windows, and export that private key to the remote server too.
From VS Code remote debug tips and tricks:
In a Powershell window, create a public/private key pair just as you would in a Linux terminal:
ssh-keygen -t rsa -b 4096
Then export it to the remote server:
export USER_AT_HOST="your-user-name-on-host@hostname"
export PUBKEYPATH="$HOME/.ssh/id_rsa.pub"
ssh $USER_AT_HOST "powershell New-Item -Force -ItemType Directory -Path \"\$HOME\\.ssh\"; Add-Content -Force -Path \"\$HOME\\.ssh\\authorized_keys\" -Value '$(tr -d '\n\r' < "$PUBKEYPATH")'"
Make sure you can connect via passwordless SSH via PowerShell.
Finally, in VS Code. press Ctrl+Shift+P to open the command palette and select "Remote-SSH: Open SSH Configuration File..." and edit the config file like so:
Host [convenient name]
HostName [hostname]
User [username]
IdentityFile C:/Users/[username]/.ssh/id_rsa*
Then when you run "Remote-SSH: Connect to Host..." in VS Code and choose the host above, it should connect without prompting for a password.