I would say that the minimum mechanical knowlege a cyclist should have depends on how far you plan to ride, whether you can get cell phone reception and whether you are in a rural or urban environment.

If you are only going to ride 5 miles out from your home, then you probably don't need much mechanical skill. If you get a flat, you should be able to walk the bike home in under 1.5 hours, or use the cell phone if you have reception, to summons help.

If you are going on a ride that will take you more than 20 miles away from home (or your starting location) then you should at least be able to fix a flat as it will take more than a few hours to walk home. Again, unless you have a cell phone and reception.

If you are going touring, then you should really be able to a least fix flats, true wheels, adjust derailleurs and brakes. In other words when touring you need to be more self-reliant.

All of the above is also dependent on where you ride. If you are driving your car to a location way out in the country to start your ride, you are already a long way from civilisation, and you should have the mechanical skills to get you back to your starting point. If you ride mostly in a place where there are plenty of bike shops, like in the city, then you only need some cash or a credit card. If you ride where bike shops are few and far between, then you need more mechanical skills.