Where to find the specifications

Listed on the furnace

Somewhere inside the service panel of the furnace, there should be a schematic for the furnace. This may list the electrical specifications of the blower motor. If not, the blower motor itself will have a label on it. It should list the voltage, amperage, horsepower, etc.

Using an Ammeter

An ammeter is used to measure the amount of current flowing through an electrical circuit. You can use a clamp on ammeter to measure the amount of current being used by the blower motor while it's running.

Simply set the meter to measure current (A in the pictured meter), and clamp it on the hot wire feeding the motor.
_{This method may require working on or near live wires, please use caution and/or consult an expert to take the readings}

Calculate Usage

With the above information and Ohm's law, it's easy to determine how much power the motor will use over a given time period.

Ohm's law says that Power (P) = Voltage (V) x Current (I), so we can determine how much power the motor uses at a single instant in time.

ex.

P = 115V * 5.2A
P = 598 Watts  

Next we have to determine how many kilowatt hours the motor will use, which we do by dividing by 1000.

P = 598 W / 1000
P = 0.598 kWh

So for every hour the motor runs, it will use 0.943 kilowatts of power.

Total power/day = 0.598kWh * 24h
Total power/day = 14.352kWh
Total power/month = 14.352Kwh * 30
Total power/month = 430.56kWh

Finally, if you multiply this number by the amount you pay per kilowatt hour. You'll see just about how much it costs to run the motor constantly.

Cost = 430.56kWh * $0.07
Cost = $30.14 per month

_{Notes:
Results using this process will be an estimate only.
Values used are not actual values, they are example values only.
Calculations may be incorrect, nobody checked my math.}

Based on the photos you provided I am sure that your heating system is hydronic, i.e. water is heated by the furnace and then flows through your baseboards giving up its heat to the room. The unit on the left in your photo is your water heater which operates independently of your heating furnace (the unit on the right). That's why you can have hot water without room heating. The large cylinder near the top of your furnace is the expansion chamber designed to handle the increase in water pressure as the water is heated by the furnace. Your problem, however, may be due to the small unit to the right of the expansion chamber, the one with the electrical conduit attached. I believe that is a damper designed to close when the furnace is off to prevent heat being wasted by going up the flue. When the furnace is on, that damper will be open to allow the gas combustion byproducts to go up the flue. If that damper is not working properly, and stays closed, the furnace is designed to turn off to prevent those byproducts from filling your basement. You should check to see if that is your problem. If your thermostat is asking for heat, and that damper is closed, then the furnace will not run and you will not get heat. One way to cure this problem is simply to disconnect the damper as it does not increase the furnace efficiency very much anyway. Or you can have it replaced. No guarantees, but this is easy enough to check. Another possibility is that the furnace pilot light is out or the thermocouple is defective. You probably will need to call a repairman for these problems.