When drag force ($bV$) equals to object's weight (mg) then upward and downward force becomes equal. As a result the object comes to rest. If this is true, how is a body moving with constant velocity?
You have the incorrect (but very common) misconception that objects that experience balanced forces will come to rest. This is just not the case. Objects that experience balanced forces continue at a constant velocity.
The root of this misconception is usually the idea that forces cause motion. That is not the case. Forces cause changes in motion. An object that is moving will continue moving forever if there are no forces (or if the forces are balanced). The reason so many people have the misconception that objects naturally come to rest (which was part of the once dominant, but now discredited, Aristotelian physics), is because in everyday life, it appears to be true. Slide a book across a table, and it seems to come to rest on its own. Roll a ball across the floor and it will eventually come to a stop. Only by continuously applying a force, does an object continue moving!
What is difficult to see, is that frictional and drag forces are everywhere! Once you realize that, you can see that objects don't naturally come to rest, the come to rest under the influences of forces, such as friction or air resistance. When you push a book across the table at a constant velocity, the reason it remains at a constant velocity is actually because your applied force, and the force of friction, are balanced.
Now that we have disassembled the misconception, let's rebuild it back into a rigorous answer to your question:
Objects move at a constant velocity, unless unbalanced forces act on it. The reason everyday objects seem to come to rest without any forces, is because it's difficult to notice the forces of friction and air resistance, but they are there. So if the forces of weight and air resistance are balanced, the object doesn't come to a stop, but continues at a constant velocity.
An object in such a state has reached what is called a dynamic equilibrium where a constant velocity, free of acceleration is occurring. This, opposed to what is called static equilibrium where the object has reached a position with zero velocity.
For a falling object that experiences body forces due to gravity ($mg$) and opposing drag force ($bV$), and where acceleration has reached zero, the dynamic equilibrium state is often called terminal velocity