When it comes to rigorous deduction, even relativists (implicitly) admit that the motion of the observer cannot alter the wavelength of the light wave:
The observer O receives a light wave from the source S. The wavelength of the emitted wave is Ls. (...) Let Ts be the time in which one wavelength is emitted as measured by a clock that is moving along with S. (...) Now let's suppose that the source is at rest and the observer is moving with velocity v in the direction of the source. Let To be the time in which the observer passes one wavelength, as measured by a clock that is moving along with the observer. In the time To the observer travels a distance v*To to the left and the light wave travels a distance Ls-v*To to the right. The light's distance is also equal to c*To.
So Ls - v*To = c*To.
Or c*Ts = c*To + v*To.
The observed period in case of a moving observer is
To = Ts(c/(c+v))
[end of quotation]
The last result, combined with the formula
(frequency) = (speed of the light wave)/(wavelength)
entails that the observer measures the frequency to be Fo=Fs(1+v/c) and the speed of the light wave to be c'=c+v.