Electricity is out, Breaker is fine

Get a cheap VOM and trace the wire to the positive terminal on the dimmer switch. If the terminal is not live you've got a bad connection. If the terminal is live, check the ground continuity. If you got a bad ground you got a bad ground connection.

If you have a good positive and a good ground the problem is with the dimmeer or the lights.

Rap

Since you are in the basement, look for a GFI circuit that has tripped. Assuming there is no GFI circuit or that it is working correctly, start at a light fixture that is not working. Turn the light switch to the on position, pull the bulb out and check for voltage in the receptacle. If you have power there, the lights should be on. Assuming you have no power there, remove the cover plate from the light switch and check for power on the inlet and outlet. If you have power on the inlet but not the outlet, the switch is the problem although it doesn't make sense why some of the outlets are out also unless the person who installed it put it upstream of the outlets. If you have no power in the inlet, double check for a GFI then double check the breaker.

As others have said check for a gfci first. If none are present but you have voltage at the light sockets when switch is on, the problem is most assuredly a broken neutral connection somewhere in the circuit.