Changes of units, in any equation, modify the coefficients. If different coefficients are subject to different scaling exponents (which is what it means to be "dimensionally wrong"), it means that for the equation to not depend on choice of units, the coefficients must be dimensionful, and in a specific way. There is a unique set of units for the coefficients making the equation dimensionally consistent, up to multiplying the entire equation by some unit, like going from A=B to A m^2 = B m^2.
Equations that accurately describe physical processes usually are not dependent on choice of units, so it is likeliest in any particular example that the coefficients have the necessary units to cancel apparent discrepancies in the dimensionality of the different terms. As the first answer states, this is true for the telegraph equation, and the principle is more general.