The following is with the presumption that you have cleared out the cabinet and carefully inspected the entire area with a flashlight and you still are not sure of the exact source of the leak.
"is this something that I can easily fix by taking apart the garbage
Garbage disposers are not considered "user serviceable". What I mean is that if you "take it apart" you will probably not be able to repair it (assuming it needs repair).
"would plumbers putty or something over the object solve the problem?"
NO, no putty or compound will ever work if you try to apply it to the areas circled in your picture.
You need to find the root cause of the water you are seeing. ISE disposers utilize an upper basket assembly that is fixed/attached to the sink as a separate piece from the main unit (main unit being the motor, housing and shredder plate compartment). The two pieces are fastened together with a locking ring. Upper piece pictured below:
Your leak could be a defect with the disposer itself. It could also be a leaky drain tube connection or a leak above at the basket assembly, and the water is running down around or through the plastic and foam sound insulating motor housing. You need to figure out which it is.
You need to disconnect the waste pipes and electrical supply (kill the power first), remove the motor/housing/shred chamber assembly and carefully inspect it.
I would reconnect the waste elbow but have it facing upwards, then fill the housing with water (while it is sitting on the floor, separated from the sink) to see if the unit is leaking. If it is, replace it. If it is not:
Remove the basket assembly from the sink by removing the steel lock-ring and backing out the the 3 mounting screws, then reassemble the whole thing using fresh plumber's putty and a new gasket, then make sure to use new slip-joint washers when reattaching the waste pipes.
If none of this sounds like it makes sense you really should call a reputable licensed plumber.