Cannot Unlock System Preferences After Upgraded to High Sierra

You need to have an account login password and be using an Admin account. If you haven't ever set a password, then you need to do that:

Forgot Your Account Password

El Capitan Password Reset.

macOS Sierra-Reset a login password.

Change or reset the password of a macOS user account - Apple Support.

I think is this how I got System Preferences to stop prompting for a password while logged-in to my administrator account. Note that my administrator account does have a password.

• Click the System Preferences icon in the Dock.
• Click the Security & Privacy icon, and then click the General tab.
• If grayed out, click the Lock icon, type an administrator name and password, and then click Unlock.
• Click Advanced.
• Uncheck the Require an administrator password to access system preferences with lock icons check box.
• Click OK.
• Click the Close button.

If the Require an administrator password to access system preferences with lock icons was set on your system, then maybe setting a password, unchecking that option, and then removing the password may resolve your problem.

That is correct. You cannot install software without a password. Fix your Admin account to use a password. It's also more secure if your computer could be accessed without your knowledge or by unwanted persons.

My administrator account has a password, but this post says a blank password might not work:

If you can't unlock settings in System Preferences - Apple Support

I think there is another System Preference setting that eliminates the prompt for a password each time you try to make a change when logged-in as an administrator. I do use that option to avoid having to provide a password each time I update a system preference as an administrator.

As I recall, it changed to always asking for a password when I upgraded to Sierra. I had to change it back to avoid the password prompt. Sorry, but I am not at may iMac and cannot recall where that option is located in System Preferences.