Can I have independent heating zones with a standard boiler?
Yes, it can be done.  It just needs to be plumbed (and wired) right.  Each thermostat should control a valve that admits hot water to its section of the house.
The boiler should be wired to fire up whenever either thermostat is calling for heat.
For example, if the basement is cold, but the rest of the house is warm, then the valve allowing water to the basement should be opened, and the boiler turned on.