A few things. Most of the time, even at supersonic speeds, Barry is talking at a normal pace. There are very few times Barry talks really fast (the one time with Iris in his police lab). In the comics and cartoons, he and other speedsters can have conversations at super speed, including the high pitch change that results from it.
A doppler shift here does not apply, for three reasons. Even while running at supersonic speeds, a closely placed microphone will be stationary, relatively speaking, to Barry, so no shift in audio will happen.
A person listening to a car's horn for example, only experience a Doppler Effect, when the source is moving relative to the person listening. If you are standing on the sidewalk with a car beeping passing by at speed, the sound will shift. If you are In the car, moving at the same speed at the same direction, you will not hear a change in pitch.
In this case, it's the microphone that's the listener, moving at the same speed Barry is moving.
And as the microphone is relatively close to Barry, this makes it less likely to experience any issues. The microphone, embedded somewhere in the suit or the comms. It may be a throat microphone, a contact microphone that doesn't even use sound in air. In any case, it is inside the air bubble that the Flash would produce at super-speed.
Barry also doesn't produce shockwaves or sonic booms. This is a very "Speed Force is magic" reason, as he should when traveling at Mach Speeds. But if he were to produce mach speed shockwaves at ground level in a dense urban city, people would be injured, and windows everywhere would be broken, or worse. So there is a super-physics reason anyway.
If Barry did produce a shockwave, the microphone would still be inside the area not affected, so communication is clear. People on a Concord super-sonic plane can talk normally, as the air inside is stationary compared to them.
The last reason is that Cisco, with the show's super-science, would have worked around the issue. Just like he did with the Suit to prevent Barry from catching on fire from running. Some pseudo technobabble later, the issue would be corrected. Or simpler, as mentioned earlier, a throat microphone would be used.
A throat mic is important though, as running outside of an enclosure, like a car/plane, will introduce a lot of wind noise, like when on a motorcycle or car with the windows down. The throat mic is mostly impervious to wind noise.