The Airspeed Indicator (ASI) measures the dynamic pressure, producing an Indicated Airspeed (IAS) Output, determined by the value of that dynamic pressure. Meaning that, every time, the ASI sense a particular value of dynamic pressure, it indicates the same IAS.
IAS or Dynamic Pressure = 1⁄2 ρV2

• This means that if we climb at constant IAS, we must climb at a constant dynamic pressure. As we climb, density decreases. In order to maintain a constant IAS => TAS must increase.

Lift = CL 1/2ρ Vsquared S
Drag = CD 1/2ρ Vsquared S
=> Both CD and CL are determined by angle of attack.

If we assume that the weight of our aircraft remains constant, then as we climb at constant IAS (which also means constant dynamic pressure) we must keep the angle of attack constant to maintain constant lift to match the constant weight. This in turn means that the CD will remain constant.

So as we climb we have a situation in which CD and 1/2ρ Vsquared are both constant, so (unless our surface area (S) changes), we must have constant drag.

So climbing at constant IAS produces constant drag.